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Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  mblfinlem4 Structured version   Visualization version   GIF version
Theorem mblfinlem4 36517
Description: Backward direction of ismblfin 36518. (Contributed by Brendan Leahy, 28-Mar-2018.) (Revised by Brendan Leahy, 13-Jul-2018.)
Assertion
Ref Expression
mblfinlem4 (((𝐴 βŠ† ℝ ∧ (vol*β€˜π΄) ∈ ℝ) ∧ 𝐴 ∈ dom vol) β†’ (vol*β€˜π΄) = sup({𝑦 ∣ βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑏 βŠ† 𝐴 ∧ 𝑦 = (volβ€˜π‘))}, ℝ, < ))
Distinct variable group:   𝑦,𝑏,𝐴
Proof of Theorem mblfinlem4
Dummy variables 𝑓 𝑔 𝑠 𝑒 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ltso 11291 . . . 4 < Or ℝ
21a1i 11 . . 3 (((𝐴 βŠ† ℝ ∧ (vol*β€˜π΄) ∈ ℝ) ∧ 𝐴 ∈ dom vol) β†’ < Or ℝ)
3 simplr 768 . . 3 (((𝐴 βŠ† ℝ ∧ (vol*β€˜π΄) ∈ ℝ) ∧ 𝐴 ∈ dom vol) β†’ (vol*β€˜π΄) ∈ ℝ)
4 vex 3479 . . . . . . 7 𝑒 ∈ V
5 eqeq1 2737 . . . . . . . . 9 (𝑦 = 𝑒 β†’ (𝑦 = (volβ€˜π‘) ↔ 𝑒 = (volβ€˜π‘)))
65anbi2d 630 . . . . . . . 8 (𝑦 = 𝑒 β†’ ((𝑏 βŠ† 𝐴 ∧ 𝑦 = (volβ€˜π‘)) ↔ (𝑏 βŠ† 𝐴 ∧ 𝑒 = (volβ€˜π‘))))
76rexbidv 3179 . . . . . . 7 (𝑦 = 𝑒 β†’ (βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑏 βŠ† 𝐴 ∧ 𝑦 = (volβ€˜π‘)) ↔ βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑏 βŠ† 𝐴 ∧ 𝑒 = (volβ€˜π‘))))
84, 7elab 3668 . . . . . 6 (𝑒 ∈ {𝑦 ∣ βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑏 βŠ† 𝐴 ∧ 𝑦 = (volβ€˜π‘))} ↔ βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑏 βŠ† 𝐴 ∧ 𝑒 = (volβ€˜π‘)))
9 simprl 770 . . . . . . . . 9 ((𝑏 ∈ (Clsdβ€˜(topGenβ€˜ran (,))) ∧ (𝑏 βŠ† 𝐴 ∧ 𝑒 = (volβ€˜π‘))) β†’ 𝑏 βŠ† 𝐴)
10 ovolss 24994 . . . . . . . . . . 11 ((𝑏 βŠ† 𝐴 ∧ 𝐴 βŠ† ℝ) β†’ (vol*β€˜π‘) ≀ (vol*β€˜π΄))
11 sstr 3990 . . . . . . . . . . . . 13 ((𝑏 βŠ† 𝐴 ∧ 𝐴 βŠ† ℝ) β†’ 𝑏 βŠ† ℝ)
12 ovolcl 24987 . . . . . . . . . . . . 13 (𝑏 βŠ† ℝ β†’ (vol*β€˜π‘) ∈ ℝ*)
1311, 12syl 17 . . . . . . . . . . . 12 ((𝑏 βŠ† 𝐴 ∧ 𝐴 βŠ† ℝ) β†’ (vol*β€˜π‘) ∈ ℝ*)
14 ovolcl 24987 . . . . . . . . . . . . 13 (𝐴 βŠ† ℝ β†’ (vol*β€˜π΄) ∈ ℝ*)
1514adantl 483 . . . . . . . . . . . 12 ((𝑏 βŠ† 𝐴 ∧ 𝐴 βŠ† ℝ) β†’ (vol*β€˜π΄) ∈ ℝ*)
16 xrlenlt 11276 . . . . . . . . . . . 12 (((vol*β€˜π‘) ∈ ℝ* ∧ (vol*β€˜π΄) ∈ ℝ*) β†’ ((vol*β€˜π‘) ≀ (vol*β€˜π΄) ↔ Β¬ (vol*β€˜π΄) < (vol*β€˜π‘)))
1713, 15, 16syl2anc 585 . . . . . . . . . . 11 ((𝑏 βŠ† 𝐴 ∧ 𝐴 βŠ† ℝ) β†’ ((vol*β€˜π‘) ≀ (vol*β€˜π΄) ↔ Β¬ (vol*β€˜π΄) < (vol*β€˜π‘)))
1810, 17mpbid 231 . . . . . . . . . 10 ((𝑏 βŠ† 𝐴 ∧ 𝐴 βŠ† ℝ) β†’ Β¬ (vol*β€˜π΄) < (vol*β€˜π‘))
1918ancoms 460 . . . . . . . . 9 ((𝐴 βŠ† ℝ ∧ 𝑏 βŠ† 𝐴) β†’ Β¬ (vol*β€˜π΄) < (vol*β€˜π‘))
209, 19sylan2 594 . . . . . . . 8 ((𝐴 βŠ† ℝ ∧ (𝑏 ∈ (Clsdβ€˜(topGenβ€˜ran (,))) ∧ (𝑏 βŠ† 𝐴 ∧ 𝑒 = (volβ€˜π‘)))) β†’ Β¬ (vol*β€˜π΄) < (vol*β€˜π‘))
21 simprrr 781 . . . . . . . . . 10 ((𝐴 βŠ† ℝ ∧ (𝑏 ∈ (Clsdβ€˜(topGenβ€˜ran (,))) ∧ (𝑏 βŠ† 𝐴 ∧ 𝑒 = (volβ€˜π‘)))) β†’ 𝑒 = (volβ€˜π‘))
22 uniretop 24271 . . . . . . . . . . . . . . 15 ℝ = βˆͺ (topGenβ€˜ran (,))
2322cldss 22525 . . . . . . . . . . . . . 14 (𝑏 ∈ (Clsdβ€˜(topGenβ€˜ran (,))) β†’ 𝑏 βŠ† ℝ)
24 dfss4 4258 . . . . . . . . . . . . . 14 (𝑏 βŠ† ℝ ↔ (ℝ βˆ– (ℝ βˆ– 𝑏)) = 𝑏)
2523, 24sylib 217 . . . . . . . . . . . . 13 (𝑏 ∈ (Clsdβ€˜(topGenβ€˜ran (,))) β†’ (ℝ βˆ– (ℝ βˆ– 𝑏)) = 𝑏)
26 rembl 25049 . . . . . . . . . . . . . 14 ℝ ∈ dom vol
2722cldopn 22527 . . . . . . . . . . . . . . 15 (𝑏 ∈ (Clsdβ€˜(topGenβ€˜ran (,))) β†’ (ℝ βˆ– 𝑏) ∈ (topGenβ€˜ran (,)))
28 opnmbl 25111 . . . . . . . . . . . . . . 15 ((ℝ βˆ– 𝑏) ∈ (topGenβ€˜ran (,)) β†’ (ℝ βˆ– 𝑏) ∈ dom vol)
2927, 28syl 17 . . . . . . . . . . . . . 14 (𝑏 ∈ (Clsdβ€˜(topGenβ€˜ran (,))) β†’ (ℝ βˆ– 𝑏) ∈ dom vol)
30 difmbl 25052 . . . . . . . . . . . . . 14 ((ℝ ∈ dom vol ∧ (ℝ βˆ– 𝑏) ∈ dom vol) β†’ (ℝ βˆ– (ℝ βˆ– 𝑏)) ∈ dom vol)
3126, 29, 30sylancr 588 . . . . . . . . . . . . 13 (𝑏 ∈ (Clsdβ€˜(topGenβ€˜ran (,))) β†’ (ℝ βˆ– (ℝ βˆ– 𝑏)) ∈ dom vol)
3225, 31eqeltrrd 2835 . . . . . . . . . . . 12 (𝑏 ∈ (Clsdβ€˜(topGenβ€˜ran (,))) β†’ 𝑏 ∈ dom vol)
33 mblvol 25039 . . . . . . . . . . . 12 (𝑏 ∈ dom vol β†’ (volβ€˜π‘) = (vol*β€˜π‘))
3432, 33syl 17 . . . . . . . . . . 11 (𝑏 ∈ (Clsdβ€˜(topGenβ€˜ran (,))) β†’ (volβ€˜π‘) = (vol*β€˜π‘))
3534ad2antrl 727 . . . . . . . . . 10 ((𝐴 βŠ† ℝ ∧ (𝑏 ∈ (Clsdβ€˜(topGenβ€˜ran (,))) ∧ (𝑏 βŠ† 𝐴 ∧ 𝑒 = (volβ€˜π‘)))) β†’ (volβ€˜π‘) = (vol*β€˜π‘))
3621, 35eqtrd 2773 . . . . . . . . 9 ((𝐴 βŠ† ℝ ∧ (𝑏 ∈ (Clsdβ€˜(topGenβ€˜ran (,))) ∧ (𝑏 βŠ† 𝐴 ∧ 𝑒 = (volβ€˜π‘)))) β†’ 𝑒 = (vol*β€˜π‘))
3736breq2d 5160 . . . . . . . 8 ((𝐴 βŠ† ℝ ∧ (𝑏 ∈ (Clsdβ€˜(topGenβ€˜ran (,))) ∧ (𝑏 βŠ† 𝐴 ∧ 𝑒 = (volβ€˜π‘)))) β†’ ((vol*β€˜π΄) < 𝑒 ↔ (vol*β€˜π΄) < (vol*β€˜π‘)))
3820, 37mtbird 325 . . . . . . 7 ((𝐴 βŠ† ℝ ∧ (𝑏 ∈ (Clsdβ€˜(topGenβ€˜ran (,))) ∧ (𝑏 βŠ† 𝐴 ∧ 𝑒 = (volβ€˜π‘)))) β†’ Β¬ (vol*β€˜π΄) < 𝑒)
3938rexlimdvaa 3157 . . . . . 6 (𝐴 βŠ† ℝ β†’ (βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑏 βŠ† 𝐴 ∧ 𝑒 = (volβ€˜π‘)) β†’ Β¬ (vol*β€˜π΄) < 𝑒))
408, 39biimtrid 241 . . . . 5 (𝐴 βŠ† ℝ β†’ (𝑒 ∈ {𝑦 ∣ βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑏 βŠ† 𝐴 ∧ 𝑦 = (volβ€˜π‘))} β†’ Β¬ (vol*β€˜π΄) < 𝑒))
4140ad2antrr 725 . . . 4 (((𝐴 βŠ† ℝ ∧ (vol*β€˜π΄) ∈ ℝ) ∧ 𝐴 ∈ dom vol) β†’ (𝑒 ∈ {𝑦 ∣ βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑏 βŠ† 𝐴 ∧ 𝑦 = (volβ€˜π‘))} β†’ Β¬ (vol*β€˜π΄) < 𝑒))
4241imp 408 . . 3 ((((𝐴 βŠ† ℝ ∧ (vol*β€˜π΄) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ 𝑒 ∈ {𝑦 ∣ βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑏 βŠ† 𝐴 ∧ 𝑦 = (volβ€˜π‘))}) β†’ Β¬ (vol*β€˜π΄) < 𝑒)
43 1rp 12975 . . . . . . . . 9 1 ∈ ℝ+
44 eqid 2733 . . . . . . . . . 10 seq1( + , ((abs ∘ βˆ’ ) ∘ 𝑓)) = seq1( + , ((abs ∘ βˆ’ ) ∘ 𝑓))
4544ovolgelb 24989 . . . . . . . . 9 ((𝐴 βŠ† ℝ ∧ (vol*β€˜π΄) ∈ ℝ ∧ 1 ∈ ℝ+) β†’ βˆƒπ‘“ ∈ (( ≀ ∩ (ℝ Γ— ℝ)) ↑m β„•)(𝐴 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ sup(ran seq1( + , ((abs ∘ βˆ’ ) ∘ 𝑓)), ℝ*, < ) ≀ ((vol*β€˜π΄) + 1)))
4643, 45mp3an3 1451 . . . . . . . 8 ((𝐴 βŠ† ℝ ∧ (vol*β€˜π΄) ∈ ℝ) β†’ βˆƒπ‘“ ∈ (( ≀ ∩ (ℝ Γ— ℝ)) ↑m β„•)(𝐴 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ sup(ran seq1( + , ((abs ∘ βˆ’ ) ∘ 𝑓)), ℝ*, < ) ≀ ((vol*β€˜π΄) + 1)))
47 elmapi 8840 . . . . . . . . . . 11 (𝑓 ∈ (( ≀ ∩ (ℝ Γ— ℝ)) ↑m β„•) β†’ 𝑓:β„•βŸΆ( ≀ ∩ (ℝ Γ— ℝ)))
48 ssid 4004 . . . . . . . . . . . . . . 15 βˆͺ ran ((,) ∘ 𝑓) βŠ† βˆͺ ran ((,) ∘ 𝑓)
4944ovollb 24988 . . . . . . . . . . . . . . 15 ((𝑓:β„•βŸΆ( ≀ ∩ (ℝ Γ— ℝ)) ∧ βˆͺ ran ((,) ∘ 𝑓) βŠ† βˆͺ ran ((,) ∘ 𝑓)) β†’ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ≀ sup(ran seq1( + , ((abs ∘ βˆ’ ) ∘ 𝑓)), ℝ*, < ))
5048, 49mpan2 690 . . . . . . . . . . . . . 14 (𝑓:β„•βŸΆ( ≀ ∩ (ℝ Γ— ℝ)) β†’ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ≀ sup(ran seq1( + , ((abs ∘ βˆ’ ) ∘ 𝑓)), ℝ*, < ))
5150adantl 483 . . . . . . . . . . . . 13 (((vol*β€˜π΄) ∈ ℝ ∧ 𝑓:β„•βŸΆ( ≀ ∩ (ℝ Γ— ℝ))) β†’ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ≀ sup(ran seq1( + , ((abs ∘ βˆ’ ) ∘ 𝑓)), ℝ*, < ))
52 eqid 2733 . . . . . . . . . . . . . . . 16 ((abs ∘ βˆ’ ) ∘ 𝑓) = ((abs ∘ βˆ’ ) ∘ 𝑓)
5352, 44ovolsf 24981 . . . . . . . . . . . . . . 15 (𝑓:β„•βŸΆ( ≀ ∩ (ℝ Γ— ℝ)) β†’ seq1( + , ((abs ∘ βˆ’ ) ∘ 𝑓)):β„•βŸΆ(0[,)+∞))
54 frn 6722 . . . . . . . . . . . . . . . 16 (seq1( + , ((abs ∘ βˆ’ ) ∘ 𝑓)):β„•βŸΆ(0[,)+∞) β†’ ran seq1( + , ((abs ∘ βˆ’ ) ∘ 𝑓)) βŠ† (0[,)+∞))
55 icossxr 13406 . . . . . . . . . . . . . . . 16 (0[,)+∞) βŠ† ℝ*
5654, 55sstrdi 3994 . . . . . . . . . . . . . . 15 (seq1( + , ((abs ∘ βˆ’ ) ∘ 𝑓)):β„•βŸΆ(0[,)+∞) β†’ ran seq1( + , ((abs ∘ βˆ’ ) ∘ 𝑓)) βŠ† ℝ*)
57 supxrcl 13291 . . . . . . . . . . . . . . 15 (ran seq1( + , ((abs ∘ βˆ’ ) ∘ 𝑓)) βŠ† ℝ* β†’ sup(ran seq1( + , ((abs ∘ βˆ’ ) ∘ 𝑓)), ℝ*, < ) ∈ ℝ*)
5853, 56, 573syl 18 . . . . . . . . . . . . . 14 (𝑓:β„•βŸΆ( ≀ ∩ (ℝ Γ— ℝ)) β†’ sup(ran seq1( + , ((abs ∘ βˆ’ ) ∘ 𝑓)), ℝ*, < ) ∈ ℝ*)
59 peano2re 11384 . . . . . . . . . . . . . . 15 ((vol*β€˜π΄) ∈ ℝ β†’ ((vol*β€˜π΄) + 1) ∈ ℝ)
6059rexrd 11261 . . . . . . . . . . . . . 14 ((vol*β€˜π΄) ∈ ℝ β†’ ((vol*β€˜π΄) + 1) ∈ ℝ*)
61 rncoss 5970 . . . . . . . . . . . . . . . . . 18 ran ((,) ∘ 𝑓) βŠ† ran (,)
6261unissi 4917 . . . . . . . . . . . . . . . . 17 βˆͺ ran ((,) ∘ 𝑓) βŠ† βˆͺ ran (,)
63 unirnioo 13423 . . . . . . . . . . . . . . . . 17 ℝ = βˆͺ ran (,)
6462, 63sseqtrri 4019 . . . . . . . . . . . . . . . 16 βˆͺ ran ((,) ∘ 𝑓) βŠ† ℝ
65 ovolcl 24987 . . . . . . . . . . . . . . . 16 (βˆͺ ran ((,) ∘ 𝑓) βŠ† ℝ β†’ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ∈ ℝ*)
6664, 65ax-mp 5 . . . . . . . . . . . . . . 15 (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ∈ ℝ*
67 xrletr 13134 . . . . . . . . . . . . . . 15 (((vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ∈ ℝ* ∧ sup(ran seq1( + , ((abs ∘ βˆ’ ) ∘ 𝑓)), ℝ*, < ) ∈ ℝ* ∧ ((vol*β€˜π΄) + 1) ∈ ℝ*) β†’ (((vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ≀ sup(ran seq1( + , ((abs ∘ βˆ’ ) ∘ 𝑓)), ℝ*, < ) ∧ sup(ran seq1( + , ((abs ∘ βˆ’ ) ∘ 𝑓)), ℝ*, < ) ≀ ((vol*β€˜π΄) + 1)) β†’ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ≀ ((vol*β€˜π΄) + 1)))
6866, 67mp3an1 1449 . . . . . . . . . . . . . 14 ((sup(ran seq1( + , ((abs ∘ βˆ’ ) ∘ 𝑓)), ℝ*, < ) ∈ ℝ* ∧ ((vol*β€˜π΄) + 1) ∈ ℝ*) β†’ (((vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ≀ sup(ran seq1( + , ((abs ∘ βˆ’ ) ∘ 𝑓)), ℝ*, < ) ∧ sup(ran seq1( + , ((abs ∘ βˆ’ ) ∘ 𝑓)), ℝ*, < ) ≀ ((vol*β€˜π΄) + 1)) β†’ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ≀ ((vol*β€˜π΄) + 1)))
6958, 60, 68syl2anr 598 . . . . . . . . . . . . 13 (((vol*β€˜π΄) ∈ ℝ ∧ 𝑓:β„•βŸΆ( ≀ ∩ (ℝ Γ— ℝ))) β†’ (((vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ≀ sup(ran seq1( + , ((abs ∘ βˆ’ ) ∘ 𝑓)), ℝ*, < ) ∧ sup(ran seq1( + , ((abs ∘ βˆ’ ) ∘ 𝑓)), ℝ*, < ) ≀ ((vol*β€˜π΄) + 1)) β†’ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ≀ ((vol*β€˜π΄) + 1)))
7051, 69mpand 694 . . . . . . . . . . . 12 (((vol*β€˜π΄) ∈ ℝ ∧ 𝑓:β„•βŸΆ( ≀ ∩ (ℝ Γ— ℝ))) β†’ (sup(ran seq1( + , ((abs ∘ βˆ’ ) ∘ 𝑓)), ℝ*, < ) ≀ ((vol*β€˜π΄) + 1) β†’ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ≀ ((vol*β€˜π΄) + 1)))
7170adantll 713 . . . . . . . . . . 11 (((𝐴 βŠ† ℝ ∧ (vol*β€˜π΄) ∈ ℝ) ∧ 𝑓:β„•βŸΆ( ≀ ∩ (ℝ Γ— ℝ))) β†’ (sup(ran seq1( + , ((abs ∘ βˆ’ ) ∘ 𝑓)), ℝ*, < ) ≀ ((vol*β€˜π΄) + 1) β†’ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ≀ ((vol*β€˜π΄) + 1)))
7247, 71sylan2 594 . . . . . . . . . 10 (((𝐴 βŠ† ℝ ∧ (vol*β€˜π΄) ∈ ℝ) ∧ 𝑓 ∈ (( ≀ ∩ (ℝ Γ— ℝ)) ↑m β„•)) β†’ (sup(ran seq1( + , ((abs ∘ βˆ’ ) ∘ 𝑓)), ℝ*, < ) ≀ ((vol*β€˜π΄) + 1) β†’ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ≀ ((vol*β€˜π΄) + 1)))
7372anim2d 613 . . . . . . . . 9 (((𝐴 βŠ† ℝ ∧ (vol*β€˜π΄) ∈ ℝ) ∧ 𝑓 ∈ (( ≀ ∩ (ℝ Γ— ℝ)) ↑m β„•)) β†’ ((𝐴 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ sup(ran seq1( + , ((abs ∘ βˆ’ ) ∘ 𝑓)), ℝ*, < ) ≀ ((vol*β€˜π΄) + 1)) β†’ (𝐴 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ≀ ((vol*β€˜π΄) + 1))))
7473reximdva 3169 . . . . . . . 8 ((𝐴 βŠ† ℝ ∧ (vol*β€˜π΄) ∈ ℝ) β†’ (βˆƒπ‘“ ∈ (( ≀ ∩ (ℝ Γ— ℝ)) ↑m β„•)(𝐴 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ sup(ran seq1( + , ((abs ∘ βˆ’ ) ∘ 𝑓)), ℝ*, < ) ≀ ((vol*β€˜π΄) + 1)) β†’ βˆƒπ‘“ ∈ (( ≀ ∩ (ℝ Γ— ℝ)) ↑m β„•)(𝐴 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ≀ ((vol*β€˜π΄) + 1))))
7546, 74mpd 15 . . . . . . 7 ((𝐴 βŠ† ℝ ∧ (vol*β€˜π΄) ∈ ℝ) β†’ βˆƒπ‘“ ∈ (( ≀ ∩ (ℝ Γ— ℝ)) ↑m β„•)(𝐴 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ≀ ((vol*β€˜π΄) + 1)))
76 rexex 3077 . . . . . . 7 (βˆƒπ‘“ ∈ (( ≀ ∩ (ℝ Γ— ℝ)) ↑m β„•)(𝐴 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ≀ ((vol*β€˜π΄) + 1)) β†’ βˆƒπ‘“(𝐴 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ≀ ((vol*β€˜π΄) + 1)))
7775, 76syl 17 . . . . . 6 ((𝐴 βŠ† ℝ ∧ (vol*β€˜π΄) ∈ ℝ) β†’ βˆƒπ‘“(𝐴 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ≀ ((vol*β€˜π΄) + 1)))
7877ad2antrr 725 . . . . 5 ((((𝐴 βŠ† ℝ ∧ (vol*β€˜π΄) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑒 ∈ ℝ ∧ 𝑒 < (vol*β€˜π΄))) β†’ βˆƒπ‘“(𝐴 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ≀ ((vol*β€˜π΄) + 1)))
79 difss 4131 . . . . . . . . . . . . . 14 (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) βŠ† βˆͺ ran ((,) ∘ 𝑓)
8079, 64sstri 3991 . . . . . . . . . . . . 13 (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) βŠ† ℝ
81 ovolcl 24987 . . . . . . . . . . . . 13 ((βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) βŠ† ℝ β†’ (vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)) ∈ ℝ*)
8280, 81ax-mp 5 . . . . . . . . . . . 12 (vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)) ∈ ℝ*
8359, 82jctil 521 . . . . . . . . . . 11 ((vol*β€˜π΄) ∈ ℝ β†’ ((vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)) ∈ ℝ* ∧ ((vol*β€˜π΄) + 1) ∈ ℝ))
8483ad4antlr 732 . . . . . . . . . 10 (((((𝐴 βŠ† ℝ ∧ (vol*β€˜π΄) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑒 ∈ ℝ ∧ 𝑒 < (vol*β€˜π΄))) ∧ (𝐴 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ≀ ((vol*β€˜π΄) + 1))) β†’ ((vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)) ∈ ℝ* ∧ ((vol*β€˜π΄) + 1) ∈ ℝ))
85 ovolss 24994 . . . . . . . . . . . . . . 15 (((βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ βˆͺ ran ((,) ∘ 𝑓) βŠ† ℝ) β†’ (vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)) ≀ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)))
8679, 64, 85mp2an 691 . . . . . . . . . . . . . 14 (vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)) ≀ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓))
87 xrletr 13134 . . . . . . . . . . . . . . . 16 (((vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)) ∈ ℝ* ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ∈ ℝ* ∧ ((vol*β€˜π΄) + 1) ∈ ℝ*) β†’ (((vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)) ≀ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ≀ ((vol*β€˜π΄) + 1)) β†’ (vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)) ≀ ((vol*β€˜π΄) + 1)))
8882, 66, 87mp3an12 1452 . . . . . . . . . . . . . . 15 (((vol*β€˜π΄) + 1) ∈ ℝ* β†’ (((vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)) ≀ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ≀ ((vol*β€˜π΄) + 1)) β†’ (vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)) ≀ ((vol*β€˜π΄) + 1)))
8960, 88syl 17 . . . . . . . . . . . . . 14 ((vol*β€˜π΄) ∈ ℝ β†’ (((vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)) ≀ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ≀ ((vol*β€˜π΄) + 1)) β†’ (vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)) ≀ ((vol*β€˜π΄) + 1)))
9086, 89mpani 695 . . . . . . . . . . . . 13 ((vol*β€˜π΄) ∈ ℝ β†’ ((vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ≀ ((vol*β€˜π΄) + 1) β†’ (vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)) ≀ ((vol*β€˜π΄) + 1)))
9190ad4antlr 732 . . . . . . . . . . . 12 (((((𝐴 βŠ† ℝ ∧ (vol*β€˜π΄) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑒 ∈ ℝ ∧ 𝑒 < (vol*β€˜π΄))) ∧ 𝐴 βŠ† βˆͺ ran ((,) ∘ 𝑓)) β†’ ((vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ≀ ((vol*β€˜π΄) + 1) β†’ (vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)) ≀ ((vol*β€˜π΄) + 1)))
9291impr 456 . . . . . . . . . . 11 (((((𝐴 βŠ† ℝ ∧ (vol*β€˜π΄) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑒 ∈ ℝ ∧ 𝑒 < (vol*β€˜π΄))) ∧ (𝐴 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ≀ ((vol*β€˜π΄) + 1))) β†’ (vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)) ≀ ((vol*β€˜π΄) + 1))
93 ovolge0 24990 . . . . . . . . . . . 12 ((βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) βŠ† ℝ β†’ 0 ≀ (vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)))
9480, 93ax-mp 5 . . . . . . . . . . 11 0 ≀ (vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴))
9592, 94jctil 521 . . . . . . . . . 10 (((((𝐴 βŠ† ℝ ∧ (vol*β€˜π΄) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑒 ∈ ℝ ∧ 𝑒 < (vol*β€˜π΄))) ∧ (𝐴 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ≀ ((vol*β€˜π΄) + 1))) β†’ (0 ≀ (vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)) ∧ (vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)) ≀ ((vol*β€˜π΄) + 1)))
96 xrrege0 13150 . . . . . . . . . 10 ((((vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)) ∈ ℝ* ∧ ((vol*β€˜π΄) + 1) ∈ ℝ) ∧ (0 ≀ (vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)) ∧ (vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)) ≀ ((vol*β€˜π΄) + 1))) β†’ (vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)) ∈ ℝ)
9784, 95, 96syl2anc 585 . . . . . . . . 9 (((((𝐴 βŠ† ℝ ∧ (vol*β€˜π΄) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑒 ∈ ℝ ∧ 𝑒 < (vol*β€˜π΄))) ∧ (𝐴 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ≀ ((vol*β€˜π΄) + 1))) β†’ (vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)) ∈ ℝ)
98 resubcl 11521 . . . . . . . . . . . . . 14 (((vol*β€˜π΄) ∈ ℝ ∧ 𝑒 ∈ ℝ) β†’ ((vol*β€˜π΄) βˆ’ 𝑒) ∈ ℝ)
9998adantrr 716 . . . . . . . . . . . . 13 (((vol*β€˜π΄) ∈ ℝ ∧ (𝑒 ∈ ℝ ∧ 𝑒 < (vol*β€˜π΄))) β†’ ((vol*β€˜π΄) βˆ’ 𝑒) ∈ ℝ)
100 posdif 11704 . . . . . . . . . . . . . . . 16 ((𝑒 ∈ ℝ ∧ (vol*β€˜π΄) ∈ ℝ) β†’ (𝑒 < (vol*β€˜π΄) ↔ 0 < ((vol*β€˜π΄) βˆ’ 𝑒)))
101100ancoms 460 . . . . . . . . . . . . . . 15 (((vol*β€˜π΄) ∈ ℝ ∧ 𝑒 ∈ ℝ) β†’ (𝑒 < (vol*β€˜π΄) ↔ 0 < ((vol*β€˜π΄) βˆ’ 𝑒)))
102101biimpd 228 . . . . . . . . . . . . . 14 (((vol*β€˜π΄) ∈ ℝ ∧ 𝑒 ∈ ℝ) β†’ (𝑒 < (vol*β€˜π΄) β†’ 0 < ((vol*β€˜π΄) βˆ’ 𝑒)))
103102impr 456 . . . . . . . . . . . . 13 (((vol*β€˜π΄) ∈ ℝ ∧ (𝑒 ∈ ℝ ∧ 𝑒 < (vol*β€˜π΄))) β†’ 0 < ((vol*β€˜π΄) βˆ’ 𝑒))
10499, 103elrpd 13010 . . . . . . . . . . . 12 (((vol*β€˜π΄) ∈ ℝ ∧ (𝑒 ∈ ℝ ∧ 𝑒 < (vol*β€˜π΄))) β†’ ((vol*β€˜π΄) βˆ’ 𝑒) ∈ ℝ+)
105104rphalfcld 13025 . . . . . . . . . . 11 (((vol*β€˜π΄) ∈ ℝ ∧ (𝑒 ∈ ℝ ∧ 𝑒 < (vol*β€˜π΄))) β†’ (((vol*β€˜π΄) βˆ’ 𝑒) / 2) ∈ ℝ+)
1063, 105sylan 581 . . . . . . . . . 10 ((((𝐴 βŠ† ℝ ∧ (vol*β€˜π΄) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑒 ∈ ℝ ∧ 𝑒 < (vol*β€˜π΄))) β†’ (((vol*β€˜π΄) βˆ’ 𝑒) / 2) ∈ ℝ+)
107106adantr 482 . . . . . . . . 9 (((((𝐴 βŠ† ℝ ∧ (vol*β€˜π΄) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑒 ∈ ℝ ∧ 𝑒 < (vol*β€˜π΄))) ∧ (𝐴 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ≀ ((vol*β€˜π΄) + 1))) β†’ (((vol*β€˜π΄) βˆ’ 𝑒) / 2) ∈ ℝ+)
108 eqid 2733 . . . . . . . . . . 11 seq1( + , ((abs ∘ βˆ’ ) ∘ 𝑔)) = seq1( + , ((abs ∘ βˆ’ ) ∘ 𝑔))
109108ovolgelb 24989 . . . . . . . . . 10 (((βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) βŠ† ℝ ∧ (vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)) ∈ ℝ ∧ (((vol*β€˜π΄) βˆ’ 𝑒) / 2) ∈ ℝ+) β†’ βˆƒπ‘” ∈ (( ≀ ∩ (ℝ Γ— ℝ)) ↑m β„•)((βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) βŠ† βˆͺ ran ((,) ∘ 𝑔) ∧ sup(ran seq1( + , ((abs ∘ βˆ’ ) ∘ 𝑔)), ℝ*, < ) ≀ ((vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)) + (((vol*β€˜π΄) βˆ’ 𝑒) / 2))))
11080, 109mp3an1 1449 . . . . . . . . 9 (((vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)) ∈ ℝ ∧ (((vol*β€˜π΄) βˆ’ 𝑒) / 2) ∈ ℝ+) β†’ βˆƒπ‘” ∈ (( ≀ ∩ (ℝ Γ— ℝ)) ↑m β„•)((βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) βŠ† βˆͺ ran ((,) ∘ 𝑔) ∧ sup(ran seq1( + , ((abs ∘ βˆ’ ) ∘ 𝑔)), ℝ*, < ) ≀ ((vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)) + (((vol*β€˜π΄) βˆ’ 𝑒) / 2))))
11197, 107, 110syl2anc 585 . . . . . . . 8 (((((𝐴 βŠ† ℝ ∧ (vol*β€˜π΄) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑒 ∈ ℝ ∧ 𝑒 < (vol*β€˜π΄))) ∧ (𝐴 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ≀ ((vol*β€˜π΄) + 1))) β†’ βˆƒπ‘” ∈ (( ≀ ∩ (ℝ Γ— ℝ)) ↑m β„•)((βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) βŠ† βˆͺ ran ((,) ∘ 𝑔) ∧ sup(ran seq1( + , ((abs ∘ βˆ’ ) ∘ 𝑔)), ℝ*, < ) ≀ ((vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)) + (((vol*β€˜π΄) βˆ’ 𝑒) / 2))))
112 elmapi 8840 . . . . . . . . . . 11 (𝑔 ∈ (( ≀ ∩ (ℝ Γ— ℝ)) ↑m β„•) β†’ 𝑔:β„•βŸΆ( ≀ ∩ (ℝ Γ— ℝ)))
113 ssid 4004 . . . . . . . . . . . . . 14 βˆͺ ran ((,) ∘ 𝑔) βŠ† βˆͺ ran ((,) ∘ 𝑔)
114108ovollb 24988 . . . . . . . . . . . . . 14 ((𝑔:β„•βŸΆ( ≀ ∩ (ℝ Γ— ℝ)) ∧ βˆͺ ran ((,) ∘ 𝑔) βŠ† βˆͺ ran ((,) ∘ 𝑔)) β†’ (vol*β€˜βˆͺ ran ((,) ∘ 𝑔)) ≀ sup(ran seq1( + , ((abs ∘ βˆ’ ) ∘ 𝑔)), ℝ*, < ))
115113, 114mpan2 690 . . . . . . . . . . . . 13 (𝑔:β„•βŸΆ( ≀ ∩ (ℝ Γ— ℝ)) β†’ (vol*β€˜βˆͺ ran ((,) ∘ 𝑔)) ≀ sup(ran seq1( + , ((abs ∘ βˆ’ ) ∘ 𝑔)), ℝ*, < ))
116115adantl 483 . . . . . . . . . . . 12 ((((((𝐴 βŠ† ℝ ∧ (vol*β€˜π΄) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑒 ∈ ℝ ∧ 𝑒 < (vol*β€˜π΄))) ∧ (𝐴 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ≀ ((vol*β€˜π΄) + 1))) ∧ 𝑔:β„•βŸΆ( ≀ ∩ (ℝ Γ— ℝ))) β†’ (vol*β€˜βˆͺ ran ((,) ∘ 𝑔)) ≀ sup(ran seq1( + , ((abs ∘ βˆ’ ) ∘ 𝑔)), ℝ*, < ))
117 eqid 2733 . . . . . . . . . . . . . . 15 ((abs ∘ βˆ’ ) ∘ 𝑔) = ((abs ∘ βˆ’ ) ∘ 𝑔)
118117, 108ovolsf 24981 . . . . . . . . . . . . . 14 (𝑔:β„•βŸΆ( ≀ ∩ (ℝ Γ— ℝ)) β†’ seq1( + , ((abs ∘ βˆ’ ) ∘ 𝑔)):β„•βŸΆ(0[,)+∞))
119 frn 6722 . . . . . . . . . . . . . . 15 (seq1( + , ((abs ∘ βˆ’ ) ∘ 𝑔)):β„•βŸΆ(0[,)+∞) β†’ ran seq1( + , ((abs ∘ βˆ’ ) ∘ 𝑔)) βŠ† (0[,)+∞))
120119, 55sstrdi 3994 . . . . . . . . . . . . . 14 (seq1( + , ((abs ∘ βˆ’ ) ∘ 𝑔)):β„•βŸΆ(0[,)+∞) β†’ ran seq1( + , ((abs ∘ βˆ’ ) ∘ 𝑔)) βŠ† ℝ*)
121 supxrcl 13291 . . . . . . . . . . . . . 14 (ran seq1( + , ((abs ∘ βˆ’ ) ∘ 𝑔)) βŠ† ℝ* β†’ sup(ran seq1( + , ((abs ∘ βˆ’ ) ∘ 𝑔)), ℝ*, < ) ∈ ℝ*)
122118, 120, 1213syl 18 . . . . . . . . . . . . 13 (𝑔:β„•βŸΆ( ≀ ∩ (ℝ Γ— ℝ)) β†’ sup(ran seq1( + , ((abs ∘ βˆ’ ) ∘ 𝑔)), ℝ*, < ) ∈ ℝ*)
12399rehalfcld 12456 . . . . . . . . . . . . . . . . 17 (((vol*β€˜π΄) ∈ ℝ ∧ (𝑒 ∈ ℝ ∧ 𝑒 < (vol*β€˜π΄))) β†’ (((vol*β€˜π΄) βˆ’ 𝑒) / 2) ∈ ℝ)
1243, 123sylan 581 . . . . . . . . . . . . . . . 16 ((((𝐴 βŠ† ℝ ∧ (vol*β€˜π΄) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑒 ∈ ℝ ∧ 𝑒 < (vol*β€˜π΄))) β†’ (((vol*β€˜π΄) βˆ’ 𝑒) / 2) ∈ ℝ)
125124adantr 482 . . . . . . . . . . . . . . 15 (((((𝐴 βŠ† ℝ ∧ (vol*β€˜π΄) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑒 ∈ ℝ ∧ 𝑒 < (vol*β€˜π΄))) ∧ (𝐴 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ≀ ((vol*β€˜π΄) + 1))) β†’ (((vol*β€˜π΄) βˆ’ 𝑒) / 2) ∈ ℝ)
12697, 125readdcld 11240 . . . . . . . . . . . . . 14 (((((𝐴 βŠ† ℝ ∧ (vol*β€˜π΄) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑒 ∈ ℝ ∧ 𝑒 < (vol*β€˜π΄))) ∧ (𝐴 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ≀ ((vol*β€˜π΄) + 1))) β†’ ((vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)) + (((vol*β€˜π΄) βˆ’ 𝑒) / 2)) ∈ ℝ)
127126rexrd 11261 . . . . . . . . . . . . 13 (((((𝐴 βŠ† ℝ ∧ (vol*β€˜π΄) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑒 ∈ ℝ ∧ 𝑒 < (vol*β€˜π΄))) ∧ (𝐴 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ≀ ((vol*β€˜π΄) + 1))) β†’ ((vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)) + (((vol*β€˜π΄) βˆ’ 𝑒) / 2)) ∈ ℝ*)
128 rncoss 5970 . . . . . . . . . . . . . . . . 17 ran ((,) ∘ 𝑔) βŠ† ran (,)
129128unissi 4917 . . . . . . . . . . . . . . . 16 βˆͺ ran ((,) ∘ 𝑔) βŠ† βˆͺ ran (,)
130129, 63sseqtrri 4019 . . . . . . . . . . . . . . 15 βˆͺ ran ((,) ∘ 𝑔) βŠ† ℝ
131 ovolcl 24987 . . . . . . . . . . . . . . 15 (βˆͺ ran ((,) ∘ 𝑔) βŠ† ℝ β†’ (vol*β€˜βˆͺ ran ((,) ∘ 𝑔)) ∈ ℝ*)
132130, 131ax-mp 5 . . . . . . . . . . . . . 14 (vol*β€˜βˆͺ ran ((,) ∘ 𝑔)) ∈ ℝ*
133 xrletr 13134 . . . . . . . . . . . . . 14 (((vol*β€˜βˆͺ ran ((,) ∘ 𝑔)) ∈ ℝ* ∧ sup(ran seq1( + , ((abs ∘ βˆ’ ) ∘ 𝑔)), ℝ*, < ) ∈ ℝ* ∧ ((vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)) + (((vol*β€˜π΄) βˆ’ 𝑒) / 2)) ∈ ℝ*) β†’ (((vol*β€˜βˆͺ ran ((,) ∘ 𝑔)) ≀ sup(ran seq1( + , ((abs ∘ βˆ’ ) ∘ 𝑔)), ℝ*, < ) ∧ sup(ran seq1( + , ((abs ∘ βˆ’ ) ∘ 𝑔)), ℝ*, < ) ≀ ((vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)) + (((vol*β€˜π΄) βˆ’ 𝑒) / 2))) β†’ (vol*β€˜βˆͺ ran ((,) ∘ 𝑔)) ≀ ((vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)) + (((vol*β€˜π΄) βˆ’ 𝑒) / 2))))
134132, 133mp3an1 1449 . . . . . . . . . . . . 13 ((sup(ran seq1( + , ((abs ∘ βˆ’ ) ∘ 𝑔)), ℝ*, < ) ∈ ℝ* ∧ ((vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)) + (((vol*β€˜π΄) βˆ’ 𝑒) / 2)) ∈ ℝ*) β†’ (((vol*β€˜βˆͺ ran ((,) ∘ 𝑔)) ≀ sup(ran seq1( + , ((abs ∘ βˆ’ ) ∘ 𝑔)), ℝ*, < ) ∧ sup(ran seq1( + , ((abs ∘ βˆ’ ) ∘ 𝑔)), ℝ*, < ) ≀ ((vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)) + (((vol*β€˜π΄) βˆ’ 𝑒) / 2))) β†’ (vol*β€˜βˆͺ ran ((,) ∘ 𝑔)) ≀ ((vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)) + (((vol*β€˜π΄) βˆ’ 𝑒) / 2))))
135122, 127, 134syl2anr 598 . . . . . . . . . . . 12 ((((((𝐴 βŠ† ℝ ∧ (vol*β€˜π΄) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑒 ∈ ℝ ∧ 𝑒 < (vol*β€˜π΄))) ∧ (𝐴 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ≀ ((vol*β€˜π΄) + 1))) ∧ 𝑔:β„•βŸΆ( ≀ ∩ (ℝ Γ— ℝ))) β†’ (((vol*β€˜βˆͺ ran ((,) ∘ 𝑔)) ≀ sup(ran seq1( + , ((abs ∘ βˆ’ ) ∘ 𝑔)), ℝ*, < ) ∧ sup(ran seq1( + , ((abs ∘ βˆ’ ) ∘ 𝑔)), ℝ*, < ) ≀ ((vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)) + (((vol*β€˜π΄) βˆ’ 𝑒) / 2))) β†’ (vol*β€˜βˆͺ ran ((,) ∘ 𝑔)) ≀ ((vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)) + (((vol*β€˜π΄) βˆ’ 𝑒) / 2))))
136116, 135mpand 694 . . . . . . . . . . 11 ((((((𝐴 βŠ† ℝ ∧ (vol*β€˜π΄) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑒 ∈ ℝ ∧ 𝑒 < (vol*β€˜π΄))) ∧ (𝐴 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ≀ ((vol*β€˜π΄) + 1))) ∧ 𝑔:β„•βŸΆ( ≀ ∩ (ℝ Γ— ℝ))) β†’ (sup(ran seq1( + , ((abs ∘ βˆ’ ) ∘ 𝑔)), ℝ*, < ) ≀ ((vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)) + (((vol*β€˜π΄) βˆ’ 𝑒) / 2)) β†’ (vol*β€˜βˆͺ ran ((,) ∘ 𝑔)) ≀ ((vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)) + (((vol*β€˜π΄) βˆ’ 𝑒) / 2))))
137112, 136sylan2 594 . . . . . . . . . 10 ((((((𝐴 βŠ† ℝ ∧ (vol*β€˜π΄) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑒 ∈ ℝ ∧ 𝑒 < (vol*β€˜π΄))) ∧ (𝐴 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ≀ ((vol*β€˜π΄) + 1))) ∧ 𝑔 ∈ (( ≀ ∩ (ℝ Γ— ℝ)) ↑m β„•)) β†’ (sup(ran seq1( + , ((abs ∘ βˆ’ ) ∘ 𝑔)), ℝ*, < ) ≀ ((vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)) + (((vol*β€˜π΄) βˆ’ 𝑒) / 2)) β†’ (vol*β€˜βˆͺ ran ((,) ∘ 𝑔)) ≀ ((vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)) + (((vol*β€˜π΄) βˆ’ 𝑒) / 2))))
138137anim2d 613 . . . . . . . . 9 ((((((𝐴 βŠ† ℝ ∧ (vol*β€˜π΄) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑒 ∈ ℝ ∧ 𝑒 < (vol*β€˜π΄))) ∧ (𝐴 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ≀ ((vol*β€˜π΄) + 1))) ∧ 𝑔 ∈ (( ≀ ∩ (ℝ Γ— ℝ)) ↑m β„•)) β†’ (((βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) βŠ† βˆͺ ran ((,) ∘ 𝑔) ∧ sup(ran seq1( + , ((abs ∘ βˆ’ ) ∘ 𝑔)), ℝ*, < ) ≀ ((vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)) + (((vol*β€˜π΄) βˆ’ 𝑒) / 2))) β†’ ((βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) βŠ† βˆͺ ran ((,) ∘ 𝑔) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑔)) ≀ ((vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)) + (((vol*β€˜π΄) βˆ’ 𝑒) / 2)))))
139138reximdva 3169 . . . . . . . 8 (((((𝐴 βŠ† ℝ ∧ (vol*β€˜π΄) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑒 ∈ ℝ ∧ 𝑒 < (vol*β€˜π΄))) ∧ (𝐴 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ≀ ((vol*β€˜π΄) + 1))) β†’ (βˆƒπ‘” ∈ (( ≀ ∩ (ℝ Γ— ℝ)) ↑m β„•)((βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) βŠ† βˆͺ ran ((,) ∘ 𝑔) ∧ sup(ran seq1( + , ((abs ∘ βˆ’ ) ∘ 𝑔)), ℝ*, < ) ≀ ((vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)) + (((vol*β€˜π΄) βˆ’ 𝑒) / 2))) β†’ βˆƒπ‘” ∈ (( ≀ ∩ (ℝ Γ— ℝ)) ↑m β„•)((βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) βŠ† βˆͺ ran ((,) ∘ 𝑔) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑔)) ≀ ((vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)) + (((vol*β€˜π΄) βˆ’ 𝑒) / 2)))))
140111, 139mpd 15 . . . . . . 7 (((((𝐴 βŠ† ℝ ∧ (vol*β€˜π΄) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑒 ∈ ℝ ∧ 𝑒 < (vol*β€˜π΄))) ∧ (𝐴 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ≀ ((vol*β€˜π΄) + 1))) β†’ βˆƒπ‘” ∈ (( ≀ ∩ (ℝ Γ— ℝ)) ↑m β„•)((βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) βŠ† βˆͺ ran ((,) ∘ 𝑔) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑔)) ≀ ((vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)) + (((vol*β€˜π΄) βˆ’ 𝑒) / 2))))
141 rexex 3077 . . . . . . 7 (βˆƒπ‘” ∈ (( ≀ ∩ (ℝ Γ— ℝ)) ↑m β„•)((βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) βŠ† βˆͺ ran ((,) ∘ 𝑔) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑔)) ≀ ((vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)) + (((vol*β€˜π΄) βˆ’ 𝑒) / 2))) β†’ βˆƒπ‘”((βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) βŠ† βˆͺ ran ((,) ∘ 𝑔) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑔)) ≀ ((vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)) + (((vol*β€˜π΄) βˆ’ 𝑒) / 2))))
142140, 141syl 17 . . . . . 6 (((((𝐴 βŠ† ℝ ∧ (vol*β€˜π΄) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑒 ∈ ℝ ∧ 𝑒 < (vol*β€˜π΄))) ∧ (𝐴 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ≀ ((vol*β€˜π΄) + 1))) β†’ βˆƒπ‘”((βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) βŠ† βˆͺ ran ((,) ∘ 𝑔) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑔)) ≀ ((vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)) + (((vol*β€˜π΄) βˆ’ 𝑒) / 2))))
14359, 66jctil 521 . . . . . . . . . . . 12 ((vol*β€˜π΄) ∈ ℝ β†’ ((vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ∈ ℝ* ∧ ((vol*β€˜π΄) + 1) ∈ ℝ))
144143ad3antlr 730 . . . . . . . . . . 11 ((((𝐴 βŠ† ℝ ∧ (vol*β€˜π΄) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑒 ∈ ℝ ∧ 𝑒 < (vol*β€˜π΄))) β†’ ((vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ∈ ℝ* ∧ ((vol*β€˜π΄) + 1) ∈ ℝ))
145 ovolge0 24990 . . . . . . . . . . . . . 14 (βˆͺ ran ((,) ∘ 𝑓) βŠ† ℝ β†’ 0 ≀ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)))
14664, 145ax-mp 5 . . . . . . . . . . . . 13 0 ≀ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓))
147146jctl 525 . . . . . . . . . . . 12 ((vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ≀ ((vol*β€˜π΄) + 1) β†’ (0 ≀ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ≀ ((vol*β€˜π΄) + 1)))
148147adantl 483 . . . . . . . . . . 11 ((𝐴 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ≀ ((vol*β€˜π΄) + 1)) β†’ (0 ≀ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ≀ ((vol*β€˜π΄) + 1)))
149 xrrege0 13150 . . . . . . . . . . 11 ((((vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ∈ ℝ* ∧ ((vol*β€˜π΄) + 1) ∈ ℝ) ∧ (0 ≀ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ≀ ((vol*β€˜π΄) + 1))) β†’ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ∈ ℝ)
150144, 148, 149syl2an 597 . . . . . . . . . 10 (((((𝐴 βŠ† ℝ ∧ (vol*β€˜π΄) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑒 ∈ ℝ ∧ 𝑒 < (vol*β€˜π΄))) ∧ (𝐴 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ≀ ((vol*β€˜π΄) + 1))) β†’ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ∈ ℝ)
151150, 125resubcld 11639 . . . . . . . . 9 (((((𝐴 βŠ† ℝ ∧ (vol*β€˜π΄) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑒 ∈ ℝ ∧ 𝑒 < (vol*β€˜π΄))) ∧ (𝐴 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ≀ ((vol*β€˜π΄) + 1))) β†’ ((vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) βˆ’ (((vol*β€˜π΄) βˆ’ 𝑒) / 2)) ∈ ℝ)
152150, 107ltsubrpd 13045 . . . . . . . . 9 (((((𝐴 βŠ† ℝ ∧ (vol*β€˜π΄) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑒 ∈ ℝ ∧ 𝑒 < (vol*β€˜π΄))) ∧ (𝐴 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ≀ ((vol*β€˜π΄) + 1))) β†’ ((vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) βˆ’ (((vol*β€˜π΄) βˆ’ 𝑒) / 2)) < (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)))
153 retop 24270 . . . . . . . . . . 11 (topGenβ€˜ran (,)) ∈ Top
154 retopbas 24269 . . . . . . . . . . . . 13 ran (,) ∈ TopBases
155 bastg 22461 . . . . . . . . . . . . 13 (ran (,) ∈ TopBases β†’ ran (,) βŠ† (topGenβ€˜ran (,)))
156154, 155ax-mp 5 . . . . . . . . . . . 12 ran (,) βŠ† (topGenβ€˜ran (,))
15761, 156sstri 3991 . . . . . . . . . . 11 ran ((,) ∘ 𝑓) βŠ† (topGenβ€˜ran (,))
158 uniopn 22391 . . . . . . . . . . 11 (((topGenβ€˜ran (,)) ∈ Top ∧ ran ((,) ∘ 𝑓) βŠ† (topGenβ€˜ran (,))) β†’ βˆͺ ran ((,) ∘ 𝑓) ∈ (topGenβ€˜ran (,)))
159153, 157, 158mp2an 691 . . . . . . . . . 10 βˆͺ ran ((,) ∘ 𝑓) ∈ (topGenβ€˜ran (,))
160 mblfinlem2 36515 . . . . . . . . . 10 ((βˆͺ ran ((,) ∘ 𝑓) ∈ (topGenβ€˜ran (,)) ∧ ((vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) βˆ’ (((vol*β€˜π΄) βˆ’ 𝑒) / 2)) ∈ ℝ ∧ ((vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) βˆ’ (((vol*β€˜π΄) βˆ’ 𝑒) / 2)) < (vol*β€˜βˆͺ ran ((,) ∘ 𝑓))) β†’ βˆƒπ‘  ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑠 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ ((vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) βˆ’ (((vol*β€˜π΄) βˆ’ 𝑒) / 2)) < (vol*β€˜π‘ )))
161159, 160mp3an1 1449 . . . . . . . . 9 ((((vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) βˆ’ (((vol*β€˜π΄) βˆ’ 𝑒) / 2)) ∈ ℝ ∧ ((vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) βˆ’ (((vol*β€˜π΄) βˆ’ 𝑒) / 2)) < (vol*β€˜βˆͺ ran ((,) ∘ 𝑓))) β†’ βˆƒπ‘  ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑠 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ ((vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) βˆ’ (((vol*β€˜π΄) βˆ’ 𝑒) / 2)) < (vol*β€˜π‘ )))
162151, 152, 161syl2anc 585 . . . . . . . 8 (((((𝐴 βŠ† ℝ ∧ (vol*β€˜π΄) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑒 ∈ ℝ ∧ 𝑒 < (vol*β€˜π΄))) ∧ (𝐴 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ≀ ((vol*β€˜π΄) + 1))) β†’ βˆƒπ‘  ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑠 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ ((vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) βˆ’ (((vol*β€˜π΄) βˆ’ 𝑒) / 2)) < (vol*β€˜π‘ )))
163162adantr 482 . . . . . . 7 ((((((𝐴 βŠ† ℝ ∧ (vol*β€˜π΄) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑒 ∈ ℝ ∧ 𝑒 < (vol*β€˜π΄))) ∧ (𝐴 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ≀ ((vol*β€˜π΄) + 1))) ∧ ((βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) βŠ† βˆͺ ran ((,) ∘ 𝑔) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑔)) ≀ ((vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)) + (((vol*β€˜π΄) βˆ’ 𝑒) / 2)))) β†’ βˆƒπ‘  ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑠 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ ((vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) βˆ’ (((vol*β€˜π΄) βˆ’ 𝑒) / 2)) < (vol*β€˜π‘ )))
164 indif2 4270 . . . . . . . . . . . . . . 15 (𝑠 ∩ (ℝ βˆ– βˆͺ ran ((,) ∘ 𝑔))) = ((𝑠 ∩ ℝ) βˆ– βˆͺ ran ((,) ∘ 𝑔))
16522cldss 22525 . . . . . . . . . . . . . . . . 17 (𝑠 ∈ (Clsdβ€˜(topGenβ€˜ran (,))) β†’ 𝑠 βŠ† ℝ)
166 df-ss 3965 . . . . . . . . . . . . . . . . 17 (𝑠 βŠ† ℝ ↔ (𝑠 ∩ ℝ) = 𝑠)
167165, 166sylib 217 . . . . . . . . . . . . . . . 16 (𝑠 ∈ (Clsdβ€˜(topGenβ€˜ran (,))) β†’ (𝑠 ∩ ℝ) = 𝑠)
168167difeq1d 4121 . . . . . . . . . . . . . . 15 (𝑠 ∈ (Clsdβ€˜(topGenβ€˜ran (,))) β†’ ((𝑠 ∩ ℝ) βˆ– βˆͺ ran ((,) ∘ 𝑔)) = (𝑠 βˆ– βˆͺ ran ((,) ∘ 𝑔)))
169164, 168eqtrid 2785 . . . . . . . . . . . . . 14 (𝑠 ∈ (Clsdβ€˜(topGenβ€˜ran (,))) β†’ (𝑠 ∩ (ℝ βˆ– βˆͺ ran ((,) ∘ 𝑔))) = (𝑠 βˆ– βˆͺ ran ((,) ∘ 𝑔)))
170128, 156sstri 3991 . . . . . . . . . . . . . . . . 17 ran ((,) ∘ 𝑔) βŠ† (topGenβ€˜ran (,))
171 uniopn 22391 . . . . . . . . . . . . . . . . 17 (((topGenβ€˜ran (,)) ∈ Top ∧ ran ((,) ∘ 𝑔) βŠ† (topGenβ€˜ran (,))) β†’ βˆͺ ran ((,) ∘ 𝑔) ∈ (topGenβ€˜ran (,)))
172153, 170, 171mp2an 691 . . . . . . . . . . . . . . . 16 βˆͺ ran ((,) ∘ 𝑔) ∈ (topGenβ€˜ran (,))
17322opncld 22529 . . . . . . . . . . . . . . . 16 (((topGenβ€˜ran (,)) ∈ Top ∧ βˆͺ ran ((,) ∘ 𝑔) ∈ (topGenβ€˜ran (,))) β†’ (ℝ βˆ– βˆͺ ran ((,) ∘ 𝑔)) ∈ (Clsdβ€˜(topGenβ€˜ran (,))))
174153, 172, 173mp2an 691 . . . . . . . . . . . . . . 15 (ℝ βˆ– βˆͺ ran ((,) ∘ 𝑔)) ∈ (Clsdβ€˜(topGenβ€˜ran (,)))
175 incld 22539 . . . . . . . . . . . . . . 15 ((𝑠 ∈ (Clsdβ€˜(topGenβ€˜ran (,))) ∧ (ℝ βˆ– βˆͺ ran ((,) ∘ 𝑔)) ∈ (Clsdβ€˜(topGenβ€˜ran (,)))) β†’ (𝑠 ∩ (ℝ βˆ– βˆͺ ran ((,) ∘ 𝑔))) ∈ (Clsdβ€˜(topGenβ€˜ran (,))))
176174, 175mpan2 690 . . . . . . . . . . . . . 14 (𝑠 ∈ (Clsdβ€˜(topGenβ€˜ran (,))) β†’ (𝑠 ∩ (ℝ βˆ– βˆͺ ran ((,) ∘ 𝑔))) ∈ (Clsdβ€˜(topGenβ€˜ran (,))))
177169, 176eqeltrrd 2835 . . . . . . . . . . . . 13 (𝑠 ∈ (Clsdβ€˜(topGenβ€˜ran (,))) β†’ (𝑠 βˆ– βˆͺ ran ((,) ∘ 𝑔)) ∈ (Clsdβ€˜(topGenβ€˜ran (,))))
178 simpr 486 . . . . . . . . . . . . . . 15 (((βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) βŠ† βˆͺ ran ((,) ∘ 𝑔) ∧ 𝑠 βŠ† βˆͺ ran ((,) ∘ 𝑓)) β†’ 𝑠 βŠ† βˆͺ ran ((,) ∘ 𝑓))
179 simpl 484 . . . . . . . . . . . . . . 15 (((βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) βŠ† βˆͺ ran ((,) ∘ 𝑔) ∧ 𝑠 βŠ† βˆͺ ran ((,) ∘ 𝑓)) β†’ (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) βŠ† βˆͺ ran ((,) ∘ 𝑔))
180178, 179ssdif2d 4143 . . . . . . . . . . . . . 14 (((βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) βŠ† βˆͺ ran ((,) ∘ 𝑔) ∧ 𝑠 βŠ† βˆͺ ran ((,) ∘ 𝑓)) β†’ (𝑠 βˆ– βˆͺ ran ((,) ∘ 𝑔)) βŠ† (βˆͺ ran ((,) ∘ 𝑓) βˆ– (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)))
181 dfin4 4267 . . . . . . . . . . . . . . 15 (βˆͺ ran ((,) ∘ 𝑓) ∩ 𝐴) = (βˆͺ ran ((,) ∘ 𝑓) βˆ– (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴))
182 inss2 4229 . . . . . . . . . . . . . . 15 (βˆͺ ran ((,) ∘ 𝑓) ∩ 𝐴) βŠ† 𝐴
183181, 182eqsstrri 4017 . . . . . . . . . . . . . 14 (βˆͺ ran ((,) ∘ 𝑓) βˆ– (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)) βŠ† 𝐴
184180, 183sstrdi 3994 . . . . . . . . . . . . 13 (((βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) βŠ† βˆͺ ran ((,) ∘ 𝑔) ∧ 𝑠 βŠ† βˆͺ ran ((,) ∘ 𝑓)) β†’ (𝑠 βˆ– βˆͺ ran ((,) ∘ 𝑔)) βŠ† 𝐴)
185 sseq1 4007 . . . . . . . . . . . . . . . 16 (𝑏 = (𝑠 βˆ– βˆͺ ran ((,) ∘ 𝑔)) β†’ (𝑏 βŠ† 𝐴 ↔ (𝑠 βˆ– βˆͺ ran ((,) ∘ 𝑔)) βŠ† 𝐴))
186185anbi1d 631 . . . . . . . . . . . . . . 15 (𝑏 = (𝑠 βˆ– βˆͺ ran ((,) ∘ 𝑔)) β†’ ((𝑏 βŠ† 𝐴 ∧ (volβ€˜(𝑠 βˆ– βˆͺ ran ((,) ∘ 𝑔))) = (volβ€˜π‘)) ↔ ((𝑠 βˆ– βˆͺ ran ((,) ∘ 𝑔)) βŠ† 𝐴 ∧ (volβ€˜(𝑠 βˆ– βˆͺ ran ((,) ∘ 𝑔))) = (volβ€˜π‘))))
187 fveq2 6889 . . . . . . . . . . . . . . . . 17 ((𝑠 βˆ– βˆͺ ran ((,) ∘ 𝑔)) = 𝑏 β†’ (volβ€˜(𝑠 βˆ– βˆͺ ran ((,) ∘ 𝑔))) = (volβ€˜π‘))
188187eqcoms 2741 . . . . . . . . . . . . . . . 16 (𝑏 = (𝑠 βˆ– βˆͺ ran ((,) ∘ 𝑔)) β†’ (volβ€˜(𝑠 βˆ– βˆͺ ran ((,) ∘ 𝑔))) = (volβ€˜π‘))
189188biantrud 533 . . . . . . . . . . . . . . 15 (𝑏 = (𝑠 βˆ– βˆͺ ran ((,) ∘ 𝑔)) β†’ ((𝑠 βˆ– βˆͺ ran ((,) ∘ 𝑔)) βŠ† 𝐴 ↔ ((𝑠 βˆ– βˆͺ ran ((,) ∘ 𝑔)) βŠ† 𝐴 ∧ (volβ€˜(𝑠 βˆ– βˆͺ ran ((,) ∘ 𝑔))) = (volβ€˜π‘))))
190186, 189bitr4d 282 . . . . . . . . . . . . . 14 (𝑏 = (𝑠 βˆ– βˆͺ ran ((,) ∘ 𝑔)) β†’ ((𝑏 βŠ† 𝐴 ∧ (volβ€˜(𝑠 βˆ– βˆͺ ran ((,) ∘ 𝑔))) = (volβ€˜π‘)) ↔ (𝑠 βˆ– βˆͺ ran ((,) ∘ 𝑔)) βŠ† 𝐴))
191190rspcev 3613 . . . . . . . . . . . . 13 (((𝑠 βˆ– βˆͺ ran ((,) ∘ 𝑔)) ∈ (Clsdβ€˜(topGenβ€˜ran (,))) ∧ (𝑠 βˆ– βˆͺ ran ((,) ∘ 𝑔)) βŠ† 𝐴) β†’ βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑏 βŠ† 𝐴 ∧ (volβ€˜(𝑠 βˆ– βˆͺ ran ((,) ∘ 𝑔))) = (volβ€˜π‘)))
192177, 184, 191syl2an 597 . . . . . . . . . . . 12 ((𝑠 ∈ (Clsdβ€˜(topGenβ€˜ran (,))) ∧ ((βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) βŠ† βˆͺ ran ((,) ∘ 𝑔) ∧ 𝑠 βŠ† βˆͺ ran ((,) ∘ 𝑓))) β†’ βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑏 βŠ† 𝐴 ∧ (volβ€˜(𝑠 βˆ– βˆͺ ran ((,) ∘ 𝑔))) = (volβ€˜π‘)))
193192an12s 648 . . . . . . . . . . 11 (((βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) βŠ† βˆͺ ran ((,) ∘ 𝑔) ∧ (𝑠 ∈ (Clsdβ€˜(topGenβ€˜ran (,))) ∧ 𝑠 βŠ† βˆͺ ran ((,) ∘ 𝑓))) β†’ βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑏 βŠ† 𝐴 ∧ (volβ€˜(𝑠 βˆ– βˆͺ ran ((,) ∘ 𝑔))) = (volβ€˜π‘)))
194193adantrrr 724 . . . . . . . . . 10 (((βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) βŠ† βˆͺ ran ((,) ∘ 𝑔) ∧ (𝑠 ∈ (Clsdβ€˜(topGenβ€˜ran (,))) ∧ (𝑠 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ ((vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) βˆ’ (((vol*β€˜π΄) βˆ’ 𝑒) / 2)) < (vol*β€˜π‘ )))) β†’ βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑏 βŠ† 𝐴 ∧ (volβ€˜(𝑠 βˆ– βˆͺ ran ((,) ∘ 𝑔))) = (volβ€˜π‘)))
195194adantlr 714 . . . . . . . . 9 ((((βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) βŠ† βˆͺ ran ((,) ∘ 𝑔) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑔)) ≀ ((vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)) + (((vol*β€˜π΄) βˆ’ 𝑒) / 2))) ∧ (𝑠 ∈ (Clsdβ€˜(topGenβ€˜ran (,))) ∧ (𝑠 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ ((vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) βˆ’ (((vol*β€˜π΄) βˆ’ 𝑒) / 2)) < (vol*β€˜π‘ )))) β†’ βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑏 βŠ† 𝐴 ∧ (volβ€˜(𝑠 βˆ– βˆͺ ran ((,) ∘ 𝑔))) = (volβ€˜π‘)))
196195adantll 713 . . . . . . . 8 (((((((𝐴 βŠ† ℝ ∧ (vol*β€˜π΄) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑒 ∈ ℝ ∧ 𝑒 < (vol*β€˜π΄))) ∧ (𝐴 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ≀ ((vol*β€˜π΄) + 1))) ∧ ((βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) βŠ† βˆͺ ran ((,) ∘ 𝑔) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑔)) ≀ ((vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)) + (((vol*β€˜π΄) βˆ’ 𝑒) / 2)))) ∧ (𝑠 ∈ (Clsdβ€˜(topGenβ€˜ran (,))) ∧ (𝑠 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ ((vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) βˆ’ (((vol*β€˜π΄) βˆ’ 𝑒) / 2)) < (vol*β€˜π‘ )))) β†’ βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑏 βŠ† 𝐴 ∧ (volβ€˜(𝑠 βˆ– βˆͺ ran ((,) ∘ 𝑔))) = (volβ€˜π‘)))
197 difss 4131 . . . . . . . . . . . 12 (𝐴 βˆ– (𝑠 βˆ– βˆͺ ran ((,) ∘ 𝑔))) βŠ† 𝐴
198 ovolsscl 24995 . . . . . . . . . . . 12 (((𝐴 βˆ– (𝑠 βˆ– βˆͺ ran ((,) ∘ 𝑔))) βŠ† 𝐴 ∧ 𝐴 βŠ† ℝ ∧ (vol*β€˜π΄) ∈ ℝ) β†’ (vol*β€˜(𝐴 βˆ– (𝑠 βˆ– βˆͺ ran ((,) ∘ 𝑔)))) ∈ ℝ)
199197, 198mp3an1 1449 . . . . . . . . . . 11 ((𝐴 βŠ† ℝ ∧ (vol*β€˜π΄) ∈ ℝ) β†’ (vol*β€˜(𝐴 βˆ– (𝑠 βˆ– βˆͺ ran ((,) ∘ 𝑔)))) ∈ ℝ)
200199ad5antr 733 . . . . . . . . . 10 (((((((𝐴 βŠ† ℝ ∧ (vol*β€˜π΄) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑒 ∈ ℝ ∧ 𝑒 < (vol*β€˜π΄))) ∧ (𝐴 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ≀ ((vol*β€˜π΄) + 1))) ∧ ((βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) βŠ† βˆͺ ran ((,) ∘ 𝑔) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑔)) ≀ ((vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)) + (((vol*β€˜π΄) βˆ’ 𝑒) / 2)))) ∧ (𝑠 ∈ (Clsdβ€˜(topGenβ€˜ran (,))) ∧ (𝑠 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ ((vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) βˆ’ (((vol*β€˜π΄) βˆ’ 𝑒) / 2)) < (vol*β€˜π‘ )))) β†’ (vol*β€˜(𝐴 βˆ– (𝑠 βˆ– βˆͺ ran ((,) ∘ 𝑔)))) ∈ ℝ)
201 simp-6r 787 . . . . . . . . . 10 (((((((𝐴 βŠ† ℝ ∧ (vol*β€˜π΄) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑒 ∈ ℝ ∧ 𝑒 < (vol*β€˜π΄))) ∧ (𝐴 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ≀ ((vol*β€˜π΄) + 1))) ∧ ((βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) βŠ† βˆͺ ran ((,) ∘ 𝑔) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑔)) ≀ ((vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)) + (((vol*β€˜π΄) βˆ’ 𝑒) / 2)))) ∧ (𝑠 ∈ (Clsdβ€˜(topGenβ€˜ran (,))) ∧ (𝑠 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ ((vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) βˆ’ (((vol*β€˜π΄) βˆ’ 𝑒) / 2)) < (vol*β€˜π‘ )))) β†’ (vol*β€˜π΄) ∈ ℝ)
202 simpl 484 . . . . . . . . . . 11 ((𝑒 ∈ ℝ ∧ 𝑒 < (vol*β€˜π΄)) β†’ 𝑒 ∈ ℝ)
203202ad4antlr 732 . . . . . . . . . 10 (((((((𝐴 βŠ† ℝ ∧ (vol*β€˜π΄) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑒 ∈ ℝ ∧ 𝑒 < (vol*β€˜π΄))) ∧ (𝐴 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ≀ ((vol*β€˜π΄) + 1))) ∧ ((βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) βŠ† βˆͺ ran ((,) ∘ 𝑔) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑔)) ≀ ((vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)) + (((vol*β€˜π΄) βˆ’ 𝑒) / 2)))) ∧ (𝑠 ∈ (Clsdβ€˜(topGenβ€˜ran (,))) ∧ (𝑠 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ ((vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) βˆ’ (((vol*β€˜π΄) βˆ’ 𝑒) / 2)) < (vol*β€˜π‘ )))) β†’ 𝑒 ∈ ℝ)
204 difdif2 4286 . . . . . . . . . . . 12 (𝐴 βˆ– (𝑠 βˆ– βˆͺ ran ((,) ∘ 𝑔))) = ((𝐴 βˆ– 𝑠) βˆͺ (𝐴 ∩ βˆͺ ran ((,) ∘ 𝑔)))
205204fveq2i 6892 . . . . . . . . . . 11 (vol*β€˜(𝐴 βˆ– (𝑠 βˆ– βˆͺ ran ((,) ∘ 𝑔)))) = (vol*β€˜((𝐴 βˆ– 𝑠) βˆͺ (𝐴 ∩ βˆͺ ran ((,) ∘ 𝑔))))
206 difss 4131 . . . . . . . . . . . . . . 15 (𝐴 βˆ– 𝑠) βŠ† 𝐴
207 inss1 4228 . . . . . . . . . . . . . . 15 (𝐴 ∩ βˆͺ ran ((,) ∘ 𝑔)) βŠ† 𝐴
208206, 207unssi 4185 . . . . . . . . . . . . . 14 ((𝐴 βˆ– 𝑠) βˆͺ (𝐴 ∩ βˆͺ ran ((,) ∘ 𝑔))) βŠ† 𝐴
209 ovolsscl 24995 . . . . . . . . . . . . . 14 ((((𝐴 βˆ– 𝑠) βˆͺ (𝐴 ∩ βˆͺ ran ((,) ∘ 𝑔))) βŠ† 𝐴 ∧ 𝐴 βŠ† ℝ ∧ (vol*β€˜π΄) ∈ ℝ) β†’ (vol*β€˜((𝐴 βˆ– 𝑠) βˆͺ (𝐴 ∩ βˆͺ ran ((,) ∘ 𝑔)))) ∈ ℝ)
210208, 209mp3an1 1449 . . . . . . . . . . . . 13 ((𝐴 βŠ† ℝ ∧ (vol*β€˜π΄) ∈ ℝ) β†’ (vol*β€˜((𝐴 βˆ– 𝑠) βˆͺ (𝐴 ∩ βˆͺ ran ((,) ∘ 𝑔)))) ∈ ℝ)
211210ad5antr 733 . . . . . . . . . . . 12 (((((((𝐴 βŠ† ℝ ∧ (vol*β€˜π΄) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑒 ∈ ℝ ∧ 𝑒 < (vol*β€˜π΄))) ∧ (𝐴 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ≀ ((vol*β€˜π΄) + 1))) ∧ ((βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) βŠ† βˆͺ ran ((,) ∘ 𝑔) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑔)) ≀ ((vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)) + (((vol*β€˜π΄) βˆ’ 𝑒) / 2)))) ∧ (𝑠 ∈ (Clsdβ€˜(topGenβ€˜ran (,))) ∧ (𝑠 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ ((vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) βˆ’ (((vol*β€˜π΄) βˆ’ 𝑒) / 2)) < (vol*β€˜π‘ )))) β†’ (vol*β€˜((𝐴 βˆ– 𝑠) βˆͺ (𝐴 ∩ βˆͺ ran ((,) ∘ 𝑔)))) ∈ ℝ)
212 ovolsscl 24995 . . . . . . . . . . . . . . 15 (((𝐴 βˆ– 𝑠) βŠ† 𝐴 ∧ 𝐴 βŠ† ℝ ∧ (vol*β€˜π΄) ∈ ℝ) β†’ (vol*β€˜(𝐴 βˆ– 𝑠)) ∈ ℝ)
213206, 212mp3an1 1449 . . . . . . . . . . . . . 14 ((𝐴 βŠ† ℝ ∧ (vol*β€˜π΄) ∈ ℝ) β†’ (vol*β€˜(𝐴 βˆ– 𝑠)) ∈ ℝ)
214213ad5antr 733 . . . . . . . . . . . . 13 (((((((𝐴 βŠ† ℝ ∧ (vol*β€˜π΄) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑒 ∈ ℝ ∧ 𝑒 < (vol*β€˜π΄))) ∧ (𝐴 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ≀ ((vol*β€˜π΄) + 1))) ∧ ((βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) βŠ† βˆͺ ran ((,) ∘ 𝑔) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑔)) ≀ ((vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)) + (((vol*β€˜π΄) βˆ’ 𝑒) / 2)))) ∧ (𝑠 ∈ (Clsdβ€˜(topGenβ€˜ran (,))) ∧ (𝑠 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ ((vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) βˆ’ (((vol*β€˜π΄) βˆ’ 𝑒) / 2)) < (vol*β€˜π‘ )))) β†’ (vol*β€˜(𝐴 βˆ– 𝑠)) ∈ ℝ)
215 ovolsscl 24995 . . . . . . . . . . . . . . 15 (((𝐴 ∩ βˆͺ ran ((,) ∘ 𝑔)) βŠ† 𝐴 ∧ 𝐴 βŠ† ℝ ∧ (vol*β€˜π΄) ∈ ℝ) β†’ (vol*β€˜(𝐴 ∩ βˆͺ ran ((,) ∘ 𝑔))) ∈ ℝ)
216207, 215mp3an1 1449 . . . . . . . . . . . . . 14 ((𝐴 βŠ† ℝ ∧ (vol*β€˜π΄) ∈ ℝ) β†’ (vol*β€˜(𝐴 ∩ βˆͺ ran ((,) ∘ 𝑔))) ∈ ℝ)
217216ad5antr 733 . . . . . . . . . . . . 13 (((((((𝐴 βŠ† ℝ ∧ (vol*β€˜π΄) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑒 ∈ ℝ ∧ 𝑒 < (vol*β€˜π΄))) ∧ (𝐴 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ≀ ((vol*β€˜π΄) + 1))) ∧ ((βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) βŠ† βˆͺ ran ((,) ∘ 𝑔) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑔)) ≀ ((vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)) + (((vol*β€˜π΄) βˆ’ 𝑒) / 2)))) ∧ (𝑠 ∈ (Clsdβ€˜(topGenβ€˜ran (,))) ∧ (𝑠 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ ((vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) βˆ’ (((vol*β€˜π΄) βˆ’ 𝑒) / 2)) < (vol*β€˜π‘ )))) β†’ (vol*β€˜(𝐴 ∩ βˆͺ ran ((,) ∘ 𝑔))) ∈ ℝ)
218214, 217readdcld 11240 . . . . . . . . . . . 12 (((((((𝐴 βŠ† ℝ ∧ (vol*β€˜π΄) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑒 ∈ ℝ ∧ 𝑒 < (vol*β€˜π΄))) ∧ (𝐴 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ≀ ((vol*β€˜π΄) + 1))) ∧ ((βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) βŠ† βˆͺ ran ((,) ∘ 𝑔) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑔)) ≀ ((vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)) + (((vol*β€˜π΄) βˆ’ 𝑒) / 2)))) ∧ (𝑠 ∈ (Clsdβ€˜(topGenβ€˜ran (,))) ∧ (𝑠 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ ((vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) βˆ’ (((vol*β€˜π΄) βˆ’ 𝑒) / 2)) < (vol*β€˜π‘ )))) β†’ ((vol*β€˜(𝐴 βˆ– 𝑠)) + (vol*β€˜(𝐴 ∩ βˆͺ ran ((,) ∘ 𝑔)))) ∈ ℝ)
2193, 202, 98syl2an 597 . . . . . . . . . . . . 13 ((((𝐴 βŠ† ℝ ∧ (vol*β€˜π΄) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑒 ∈ ℝ ∧ 𝑒 < (vol*β€˜π΄))) β†’ ((vol*β€˜π΄) βˆ’ 𝑒) ∈ ℝ)
220219ad3antrrr 729 . . . . . . . . . . . 12 (((((((𝐴 βŠ† ℝ ∧ (vol*β€˜π΄) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑒 ∈ ℝ ∧ 𝑒 < (vol*β€˜π΄))) ∧ (𝐴 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ≀ ((vol*β€˜π΄) + 1))) ∧ ((βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) βŠ† βˆͺ ran ((,) ∘ 𝑔) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑔)) ≀ ((vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)) + (((vol*β€˜π΄) βˆ’ 𝑒) / 2)))) ∧ (𝑠 ∈ (Clsdβ€˜(topGenβ€˜ran (,))) ∧ (𝑠 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ ((vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) βˆ’ (((vol*β€˜π΄) βˆ’ 𝑒) / 2)) < (vol*β€˜π‘ )))) β†’ ((vol*β€˜π΄) βˆ’ 𝑒) ∈ ℝ)
221 ssdifss 4135 . . . . . . . . . . . . . . 15 (𝐴 βŠ† ℝ β†’ (𝐴 βˆ– 𝑠) βŠ† ℝ)
222221adantr 482 . . . . . . . . . . . . . 14 ((𝐴 βŠ† ℝ ∧ (vol*β€˜π΄) ∈ ℝ) β†’ (𝐴 βˆ– 𝑠) βŠ† ℝ)
223 ssinss1 4237 . . . . . . . . . . . . . . 15 (𝐴 βŠ† ℝ β†’ (𝐴 ∩ βˆͺ ran ((,) ∘ 𝑔)) βŠ† ℝ)
224223adantr 482 . . . . . . . . . . . . . 14 ((𝐴 βŠ† ℝ ∧ (vol*β€˜π΄) ∈ ℝ) β†’ (𝐴 ∩ βˆͺ ran ((,) ∘ 𝑔)) βŠ† ℝ)
225 ovolun 25008 . . . . . . . . . . . . . 14 ((((𝐴 βˆ– 𝑠) βŠ† ℝ ∧ (vol*β€˜(𝐴 βˆ– 𝑠)) ∈ ℝ) ∧ ((𝐴 ∩ βˆͺ ran ((,) ∘ 𝑔)) βŠ† ℝ ∧ (vol*β€˜(𝐴 ∩ βˆͺ ran ((,) ∘ 𝑔))) ∈ ℝ)) β†’ (vol*β€˜((𝐴 βˆ– 𝑠) βˆͺ (𝐴 ∩ βˆͺ ran ((,) ∘ 𝑔)))) ≀ ((vol*β€˜(𝐴 βˆ– 𝑠)) + (vol*β€˜(𝐴 ∩ βˆͺ ran ((,) ∘ 𝑔)))))
226222, 213, 224, 216, 225syl22anc 838 . . . . . . . . . . . . 13 ((𝐴 βŠ† ℝ ∧ (vol*β€˜π΄) ∈ ℝ) β†’ (vol*β€˜((𝐴 βˆ– 𝑠) βˆͺ (𝐴 ∩ βˆͺ ran ((,) ∘ 𝑔)))) ≀ ((vol*β€˜(𝐴 βˆ– 𝑠)) + (vol*β€˜(𝐴 ∩ βˆͺ ran ((,) ∘ 𝑔)))))
227226ad5antr 733 . . . . . . . . . . . 12 (((((((𝐴 βŠ† ℝ ∧ (vol*β€˜π΄) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑒 ∈ ℝ ∧ 𝑒 < (vol*β€˜π΄))) ∧ (𝐴 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ≀ ((vol*β€˜π΄) + 1))) ∧ ((βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) βŠ† βˆͺ ran ((,) ∘ 𝑔) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑔)) ≀ ((vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)) + (((vol*β€˜π΄) βˆ’ 𝑒) / 2)))) ∧ (𝑠 ∈ (Clsdβ€˜(topGenβ€˜ran (,))) ∧ (𝑠 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ ((vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) βˆ’ (((vol*β€˜π΄) βˆ’ 𝑒) / 2)) < (vol*β€˜π‘ )))) β†’ (vol*β€˜((𝐴 βˆ– 𝑠) βˆͺ (𝐴 ∩ βˆͺ ran ((,) ∘ 𝑔)))) ≀ ((vol*β€˜(𝐴 βˆ– 𝑠)) + (vol*β€˜(𝐴 ∩ βˆͺ ran ((,) ∘ 𝑔)))))
228124ad2antrr 725 . . . . . . . . . . . . . . 15 ((((((𝐴 βŠ† ℝ ∧ (vol*β€˜π΄) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑒 ∈ ℝ ∧ 𝑒 < (vol*β€˜π΄))) ∧ (𝐴 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ≀ ((vol*β€˜π΄) + 1))) ∧ ((βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) βŠ† βˆͺ ran ((,) ∘ 𝑔) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑔)) ≀ ((vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)) + (((vol*β€˜π΄) βˆ’ 𝑒) / 2)))) β†’ (((vol*β€˜π΄) βˆ’ 𝑒) / 2) ∈ ℝ)
229228adantr 482 . . . . . . . . . . . . . 14 (((((((𝐴 βŠ† ℝ ∧ (vol*β€˜π΄) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑒 ∈ ℝ ∧ 𝑒 < (vol*β€˜π΄))) ∧ (𝐴 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ≀ ((vol*β€˜π΄) + 1))) ∧ ((βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) βŠ† βˆͺ ran ((,) ∘ 𝑔) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑔)) ≀ ((vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)) + (((vol*β€˜π΄) βˆ’ 𝑒) / 2)))) ∧ (𝑠 ∈ (Clsdβ€˜(topGenβ€˜ran (,))) ∧ (𝑠 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ ((vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) βˆ’ (((vol*β€˜π΄) βˆ’ 𝑒) / 2)) < (vol*β€˜π‘ )))) β†’ (((vol*β€˜π΄) βˆ’ 𝑒) / 2) ∈ ℝ)
230150ad2antrr 725 . . . . . . . . . . . . . . . 16 (((((((𝐴 βŠ† ℝ ∧ (vol*β€˜π΄) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑒 ∈ ℝ ∧ 𝑒 < (vol*β€˜π΄))) ∧ (𝐴 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ≀ ((vol*β€˜π΄) + 1))) ∧ ((βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) βŠ† βˆͺ ran ((,) ∘ 𝑔) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑔)) ≀ ((vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)) + (((vol*β€˜π΄) βˆ’ 𝑒) / 2)))) ∧ (𝑠 ∈ (Clsdβ€˜(topGenβ€˜ran (,))) ∧ (𝑠 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ ((vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) βˆ’ (((vol*β€˜π΄) βˆ’ 𝑒) / 2)) < (vol*β€˜π‘ )))) β†’ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ∈ ℝ)
231 simprl 770 . . . . . . . . . . . . . . . . 17 ((𝑠 ∈ (Clsdβ€˜(topGenβ€˜ran (,))) ∧ (𝑠 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ ((vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) βˆ’ (((vol*β€˜π΄) βˆ’ 𝑒) / 2)) < (vol*β€˜π‘ ))) β†’ 𝑠 βŠ† βˆͺ ran ((,) ∘ 𝑓))
232150adantr 482 . . . . . . . . . . . . . . . . 17 ((((((𝐴 βŠ† ℝ ∧ (vol*β€˜π΄) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑒 ∈ ℝ ∧ 𝑒 < (vol*β€˜π΄))) ∧ (𝐴 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ≀ ((vol*β€˜π΄) + 1))) ∧ ((βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) βŠ† βˆͺ ran ((,) ∘ 𝑔) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑔)) ≀ ((vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)) + (((vol*β€˜π΄) βˆ’ 𝑒) / 2)))) β†’ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ∈ ℝ)
233 ovolsscl 24995 . . . . . . . . . . . . . . . . . 18 ((𝑠 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ βˆͺ ran ((,) ∘ 𝑓) βŠ† ℝ ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ∈ ℝ) β†’ (vol*β€˜π‘ ) ∈ ℝ)
23464, 233mp3an2 1450 . . . . . . . . . . . . . . . . 17 ((𝑠 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ∈ ℝ) β†’ (vol*β€˜π‘ ) ∈ ℝ)
235231, 232, 234syl2anr 598 . . . . . . . . . . . . . . . 16 (((((((𝐴 βŠ† ℝ ∧ (vol*β€˜π΄) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑒 ∈ ℝ ∧ 𝑒 < (vol*β€˜π΄))) ∧ (𝐴 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ≀ ((vol*β€˜π΄) + 1))) ∧ ((βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) βŠ† βˆͺ ran ((,) ∘ 𝑔) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑔)) ≀ ((vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)) + (((vol*β€˜π΄) βˆ’ 𝑒) / 2)))) ∧ (𝑠 ∈ (Clsdβ€˜(topGenβ€˜ran (,))) ∧ (𝑠 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ ((vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) βˆ’ (((vol*β€˜π΄) βˆ’ 𝑒) / 2)) < (vol*β€˜π‘ )))) β†’ (vol*β€˜π‘ ) ∈ ℝ)
236230, 235resubcld 11639 . . . . . . . . . . . . . . 15 (((((((𝐴 βŠ† ℝ ∧ (vol*β€˜π΄) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑒 ∈ ℝ ∧ 𝑒 < (vol*β€˜π΄))) ∧ (𝐴 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ≀ ((vol*β€˜π΄) + 1))) ∧ ((βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) βŠ† βˆͺ ran ((,) ∘ 𝑔) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑔)) ≀ ((vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)) + (((vol*β€˜π΄) βˆ’ 𝑒) / 2)))) ∧ (𝑠 ∈ (Clsdβ€˜(topGenβ€˜ran (,))) ∧ (𝑠 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ ((vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) βˆ’ (((vol*β€˜π΄) βˆ’ 𝑒) / 2)) < (vol*β€˜π‘ )))) β†’ ((vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) βˆ’ (vol*β€˜π‘ )) ∈ ℝ)
237 ssdif 4139 . . . . . . . . . . . . . . . . . . 19 (𝐴 βŠ† βˆͺ ran ((,) ∘ 𝑓) β†’ (𝐴 βˆ– 𝑠) βŠ† (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝑠))
238 difss 4131 . . . . . . . . . . . . . . . . . . . 20 (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝑠) βŠ† βˆͺ ran ((,) ∘ 𝑓)
239238, 64sstri 3991 . . . . . . . . . . . . . . . . . . 19 (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝑠) βŠ† ℝ
240 ovolss 24994 . . . . . . . . . . . . . . . . . . 19 (((𝐴 βˆ– 𝑠) βŠ† (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝑠) ∧ (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝑠) βŠ† ℝ) β†’ (vol*β€˜(𝐴 βˆ– 𝑠)) ≀ (vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝑠)))
241237, 239, 240sylancl 587 . . . . . . . . . . . . . . . . . 18 (𝐴 βŠ† βˆͺ ran ((,) ∘ 𝑓) β†’ (vol*β€˜(𝐴 βˆ– 𝑠)) ≀ (vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝑠)))
242241adantr 482 . . . . . . . . . . . . . . . . 17 ((𝐴 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ≀ ((vol*β€˜π΄) + 1)) β†’ (vol*β€˜(𝐴 βˆ– 𝑠)) ≀ (vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝑠)))
243242ad3antlr 730 . . . . . . . . . . . . . . . 16 (((((((𝐴 βŠ† ℝ ∧ (vol*β€˜π΄) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑒 ∈ ℝ ∧ 𝑒 < (vol*β€˜π΄))) ∧ (𝐴 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ≀ ((vol*β€˜π΄) + 1))) ∧ ((βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) βŠ† βˆͺ ran ((,) ∘ 𝑔) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑔)) ≀ ((vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)) + (((vol*β€˜π΄) βˆ’ 𝑒) / 2)))) ∧ (𝑠 ∈ (Clsdβ€˜(topGenβ€˜ran (,))) ∧ (𝑠 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ ((vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) βˆ’ (((vol*β€˜π΄) βˆ’ 𝑒) / 2)) < (vol*β€˜π‘ )))) β†’ (vol*β€˜(𝐴 βˆ– 𝑠)) ≀ (vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝑠)))
244 eleq1w 2817 . . . . . . . . . . . . . . . . . . . . . 22 (𝑏 = 𝑠 β†’ (𝑏 ∈ dom vol ↔ 𝑠 ∈ dom vol))
245244, 32vtoclga 3566 . . . . . . . . . . . . . . . . . . . . 21 (𝑠 ∈ (Clsdβ€˜(topGenβ€˜ran (,))) β†’ 𝑠 ∈ dom vol)
246245adantr 482 . . . . . . . . . . . . . . . . . . . 20 ((𝑠 ∈ (Clsdβ€˜(topGenβ€˜ran (,))) ∧ (𝑠 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ ((vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) βˆ’ (((vol*β€˜π΄) βˆ’ 𝑒) / 2)) < (vol*β€˜π‘ ))) β†’ 𝑠 ∈ dom vol)
247 mblsplit 25041 . . . . . . . . . . . . . . . . . . . . 21 ((𝑠 ∈ dom vol ∧ βˆͺ ran ((,) ∘ 𝑓) βŠ† ℝ ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ∈ ℝ) β†’ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) = ((vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) ∩ 𝑠)) + (vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝑠))))
24864, 247mp3an2 1450 . . . . . . . . . . . . . . . . . . . 20 ((𝑠 ∈ dom vol ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ∈ ℝ) β†’ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) = ((vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) ∩ 𝑠)) + (vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝑠))))
249246, 232, 248syl2anr 598 . . . . . . . . . . . . . . . . . . 19 (((((((𝐴 βŠ† ℝ ∧ (vol*β€˜π΄) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑒 ∈ ℝ ∧ 𝑒 < (vol*β€˜π΄))) ∧ (𝐴 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ≀ ((vol*β€˜π΄) + 1))) ∧ ((βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) βŠ† βˆͺ ran ((,) ∘ 𝑔) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑔)) ≀ ((vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)) + (((vol*β€˜π΄) βˆ’ 𝑒) / 2)))) ∧ (𝑠 ∈ (Clsdβ€˜(topGenβ€˜ran (,))) ∧ (𝑠 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ ((vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) βˆ’ (((vol*β€˜π΄) βˆ’ 𝑒) / 2)) < (vol*β€˜π‘ )))) β†’ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) = ((vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) ∩ 𝑠)) + (vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝑠))))
250249eqcomd 2739 . . . . . . . . . . . . . . . . . 18 (((((((𝐴 βŠ† ℝ ∧ (vol*β€˜π΄) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑒 ∈ ℝ ∧ 𝑒 < (vol*β€˜π΄))) ∧ (𝐴 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ≀ ((vol*β€˜π΄) + 1))) ∧ ((βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) βŠ† βˆͺ ran ((,) ∘ 𝑔) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑔)) ≀ ((vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)) + (((vol*β€˜π΄) βˆ’ 𝑒) / 2)))) ∧ (𝑠 ∈ (Clsdβ€˜(topGenβ€˜ran (,))) ∧ (𝑠 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ ((vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) βˆ’ (((vol*β€˜π΄) βˆ’ 𝑒) / 2)) < (vol*β€˜π‘ )))) β†’ ((vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) ∩ 𝑠)) + (vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝑠))) = (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)))
251230recnd 11239 . . . . . . . . . . . . . . . . . . 19 (((((((𝐴 βŠ† ℝ ∧ (vol*β€˜π΄) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑒 ∈ ℝ ∧ 𝑒 < (vol*β€˜π΄))) ∧ (𝐴 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ≀ ((vol*β€˜π΄) + 1))) ∧ ((βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) βŠ† βˆͺ ran ((,) ∘ 𝑔) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑔)) ≀ ((vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)) + (((vol*β€˜π΄) βˆ’ 𝑒) / 2)))) ∧ (𝑠 ∈ (Clsdβ€˜(topGenβ€˜ran (,))) ∧ (𝑠 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ ((vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) βˆ’ (((vol*β€˜π΄) βˆ’ 𝑒) / 2)) < (vol*β€˜π‘ )))) β†’ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ∈ β„‚)
252 inss1 4228 . . . . . . . . . . . . . . . . . . . . . . 23 (βˆͺ ran ((,) ∘ 𝑓) ∩ 𝑠) βŠ† βˆͺ ran ((,) ∘ 𝑓)
253 ovolsscl 24995 . . . . . . . . . . . . . . . . . . . . . . 23 (((βˆͺ ran ((,) ∘ 𝑓) ∩ 𝑠) βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ βˆͺ ran ((,) ∘ 𝑓) βŠ† ℝ ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ∈ ℝ) β†’ (vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) ∩ 𝑠)) ∈ ℝ)
254252, 64, 253mp3an12 1452 . . . . . . . . . . . . . . . . . . . . . 22 ((vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ∈ ℝ β†’ (vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) ∩ 𝑠)) ∈ ℝ)
255150, 254syl 17 . . . . . . . . . . . . . . . . . . . . 21 (((((𝐴 βŠ† ℝ ∧ (vol*β€˜π΄) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑒 ∈ ℝ ∧ 𝑒 < (vol*β€˜π΄))) ∧ (𝐴 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ≀ ((vol*β€˜π΄) + 1))) β†’ (vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) ∩ 𝑠)) ∈ ℝ)
256255ad2antrr 725 . . . . . . . . . . . . . . . . . . . 20 (((((((𝐴 βŠ† ℝ ∧ (vol*β€˜π΄) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑒 ∈ ℝ ∧ 𝑒 < (vol*β€˜π΄))) ∧ (𝐴 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ≀ ((vol*β€˜π΄) + 1))) ∧ ((βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) βŠ† βˆͺ ran ((,) ∘ 𝑔) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑔)) ≀ ((vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)) + (((vol*β€˜π΄) βˆ’ 𝑒) / 2)))) ∧ (𝑠 ∈ (Clsdβ€˜(topGenβ€˜ran (,))) ∧ (𝑠 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ ((vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) βˆ’ (((vol*β€˜π΄) βˆ’ 𝑒) / 2)) < (vol*β€˜π‘ )))) β†’ (vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) ∩ 𝑠)) ∈ ℝ)
257256recnd 11239 . . . . . . . . . . . . . . . . . . 19 (((((((𝐴 βŠ† ℝ ∧ (vol*β€˜π΄) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑒 ∈ ℝ ∧ 𝑒 < (vol*β€˜π΄))) ∧ (𝐴 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ≀ ((vol*β€˜π΄) + 1))) ∧ ((βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) βŠ† βˆͺ ran ((,) ∘ 𝑔) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑔)) ≀ ((vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)) + (((vol*β€˜π΄) βˆ’ 𝑒) / 2)))) ∧ (𝑠 ∈ (Clsdβ€˜(topGenβ€˜ran (,))) ∧ (𝑠 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ ((vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) βˆ’ (((vol*β€˜π΄) βˆ’ 𝑒) / 2)) < (vol*β€˜π‘ )))) β†’ (vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) ∩ 𝑠)) ∈ β„‚)
258 ovolsscl 24995 . . . . . . . . . . . . . . . . . . . . . . 23 (((βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝑠) βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ βˆͺ ran ((,) ∘ 𝑓) βŠ† ℝ ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ∈ ℝ) β†’ (vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝑠)) ∈ ℝ)
259238, 64, 258mp3an12 1452 . . . . . . . . . . . . . . . . . . . . . 22 ((vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ∈ ℝ β†’ (vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝑠)) ∈ ℝ)
260150, 259syl 17 . . . . . . . . . . . . . . . . . . . . 21 (((((𝐴 βŠ† ℝ ∧ (vol*β€˜π΄) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑒 ∈ ℝ ∧ 𝑒 < (vol*β€˜π΄))) ∧ (𝐴 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ≀ ((vol*β€˜π΄) + 1))) β†’ (vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝑠)) ∈ ℝ)
261260recnd 11239 . . . . . . . . . . . . . . . . . . . 20 (((((𝐴 βŠ† ℝ ∧ (vol*β€˜π΄) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑒 ∈ ℝ ∧ 𝑒 < (vol*β€˜π΄))) ∧ (𝐴 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ≀ ((vol*β€˜π΄) + 1))) β†’ (vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝑠)) ∈ β„‚)
262261ad2antrr 725 . . . . . . . . . . . . . . . . . . 19 (((((((𝐴 βŠ† ℝ ∧ (vol*β€˜π΄) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑒 ∈ ℝ ∧ 𝑒 < (vol*β€˜π΄))) ∧ (𝐴 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ≀ ((vol*β€˜π΄) + 1))) ∧ ((βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) βŠ† βˆͺ ran ((,) ∘ 𝑔) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑔)) ≀ ((vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)) + (((vol*β€˜π΄) βˆ’ 𝑒) / 2)))) ∧ (𝑠 ∈ (Clsdβ€˜(topGenβ€˜ran (,))) ∧ (𝑠 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ ((vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) βˆ’ (((vol*β€˜π΄) βˆ’ 𝑒) / 2)) < (vol*β€˜π‘ )))) β†’ (vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝑠)) ∈ β„‚)
263251, 257, 262subaddd 11586 . . . . . . . . . . . . . . . . . 18 (((((((𝐴 βŠ† ℝ ∧ (vol*β€˜π΄) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑒 ∈ ℝ ∧ 𝑒 < (vol*β€˜π΄))) ∧ (𝐴 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ≀ ((vol*β€˜π΄) + 1))) ∧ ((βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) βŠ† βˆͺ ran ((,) ∘ 𝑔) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑔)) ≀ ((vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)) + (((vol*β€˜π΄) βˆ’ 𝑒) / 2)))) ∧ (𝑠 ∈ (Clsdβ€˜(topGenβ€˜ran (,))) ∧ (𝑠 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ ((vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) βˆ’ (((vol*β€˜π΄) βˆ’ 𝑒) / 2)) < (vol*β€˜π‘ )))) β†’ (((vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) βˆ’ (vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) ∩ 𝑠))) = (vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝑠)) ↔ ((vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) ∩ 𝑠)) + (vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝑠))) = (vol*β€˜βˆͺ ran ((,) ∘ 𝑓))))
264250, 263mpbird 257 . . . . . . . . . . . . . . . . 17 (((((((𝐴 βŠ† ℝ ∧ (vol*β€˜π΄) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑒 ∈ ℝ ∧ 𝑒 < (vol*β€˜π΄))) ∧ (𝐴 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ≀ ((vol*β€˜π΄) + 1))) ∧ ((βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) βŠ† βˆͺ ran ((,) ∘ 𝑔) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑔)) ≀ ((vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)) + (((vol*β€˜π΄) βˆ’ 𝑒) / 2)))) ∧ (𝑠 ∈ (Clsdβ€˜(topGenβ€˜ran (,))) ∧ (𝑠 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ ((vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) βˆ’ (((vol*β€˜π΄) βˆ’ 𝑒) / 2)) < (vol*β€˜π‘ )))) β†’ ((vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) βˆ’ (vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) ∩ 𝑠))) = (vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝑠)))
265 sseqin2 4215 . . . . . . . . . . . . . . . . . . . . . 22 (𝑠 βŠ† βˆͺ ran ((,) ∘ 𝑓) ↔ (βˆͺ ran ((,) ∘ 𝑓) ∩ 𝑠) = 𝑠)
266265biimpi 215 . . . . . . . . . . . . . . . . . . . . 21 (𝑠 βŠ† βˆͺ ran ((,) ∘ 𝑓) β†’ (βˆͺ ran ((,) ∘ 𝑓) ∩ 𝑠) = 𝑠)
267266fveq2d 6893 . . . . . . . . . . . . . . . . . . . 20 (𝑠 βŠ† βˆͺ ran ((,) ∘ 𝑓) β†’ (vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) ∩ 𝑠)) = (vol*β€˜π‘ ))
268267oveq2d 7422 . . . . . . . . . . . . . . . . . . 19 (𝑠 βŠ† βˆͺ ran ((,) ∘ 𝑓) β†’ ((vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) βˆ’ (vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) ∩ 𝑠))) = ((vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) βˆ’ (vol*β€˜π‘ )))
269268adantr 482 . . . . . . . . . . . . . . . . . 18 ((𝑠 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ ((vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) βˆ’ (((vol*β€˜π΄) βˆ’ 𝑒) / 2)) < (vol*β€˜π‘ )) β†’ ((vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) βˆ’ (vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) ∩ 𝑠))) = ((vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) βˆ’ (vol*β€˜π‘ )))
270269ad2antll 728 . . . . . . . . . . . . . . . . 17 (((((((𝐴 βŠ† ℝ ∧ (vol*β€˜π΄) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑒 ∈ ℝ ∧ 𝑒 < (vol*β€˜π΄))) ∧ (𝐴 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ≀ ((vol*β€˜π΄) + 1))) ∧ ((βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) βŠ† βˆͺ ran ((,) ∘ 𝑔) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑔)) ≀ ((vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)) + (((vol*β€˜π΄) βˆ’ 𝑒) / 2)))) ∧ (𝑠 ∈ (Clsdβ€˜(topGenβ€˜ran (,))) ∧ (𝑠 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ ((vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) βˆ’ (((vol*β€˜π΄) βˆ’ 𝑒) / 2)) < (vol*β€˜π‘ )))) β†’ ((vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) βˆ’ (vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) ∩ 𝑠))) = ((vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) βˆ’ (vol*β€˜π‘ )))
271264, 270eqtr3d 2775 . . . . . . . . . . . . . . . 16 (((((((𝐴 βŠ† ℝ ∧ (vol*β€˜π΄) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑒 ∈ ℝ ∧ 𝑒 < (vol*β€˜π΄))) ∧ (𝐴 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ≀ ((vol*β€˜π΄) + 1))) ∧ ((βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) βŠ† βˆͺ ran ((,) ∘ 𝑔) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑔)) ≀ ((vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)) + (((vol*β€˜π΄) βˆ’ 𝑒) / 2)))) ∧ (𝑠 ∈ (Clsdβ€˜(topGenβ€˜ran (,))) ∧ (𝑠 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ ((vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) βˆ’ (((vol*β€˜π΄) βˆ’ 𝑒) / 2)) < (vol*β€˜π‘ )))) β†’ (vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝑠)) = ((vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) βˆ’ (vol*β€˜π‘ )))
272243, 271breqtrd 5174 . . . . . . . . . . . . . . 15 (((((((𝐴 βŠ† ℝ ∧ (vol*β€˜π΄) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑒 ∈ ℝ ∧ 𝑒 < (vol*β€˜π΄))) ∧ (𝐴 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ≀ ((vol*β€˜π΄) + 1))) ∧ ((βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) βŠ† βˆͺ ran ((,) ∘ 𝑔) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑔)) ≀ ((vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)) + (((vol*β€˜π΄) βˆ’ 𝑒) / 2)))) ∧ (𝑠 ∈ (Clsdβ€˜(topGenβ€˜ran (,))) ∧ (𝑠 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ ((vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) βˆ’ (((vol*β€˜π΄) βˆ’ 𝑒) / 2)) < (vol*β€˜π‘ )))) β†’ (vol*β€˜(𝐴 βˆ– 𝑠)) ≀ ((vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) βˆ’ (vol*β€˜π‘ )))
273 simprrr 781 . . . . . . . . . . . . . . . 16 (((((((𝐴 βŠ† ℝ ∧ (vol*β€˜π΄) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑒 ∈ ℝ ∧ 𝑒 < (vol*β€˜π΄))) ∧ (𝐴 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ≀ ((vol*β€˜π΄) + 1))) ∧ ((βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) βŠ† βˆͺ ran ((,) ∘ 𝑔) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑔)) ≀ ((vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)) + (((vol*β€˜π΄) βˆ’ 𝑒) / 2)))) ∧ (𝑠 ∈ (Clsdβ€˜(topGenβ€˜ran (,))) ∧ (𝑠 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ ((vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) βˆ’ (((vol*β€˜π΄) βˆ’ 𝑒) / 2)) < (vol*β€˜π‘ )))) β†’ ((vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) βˆ’ (((vol*β€˜π΄) βˆ’ 𝑒) / 2)) < (vol*β€˜π‘ ))
274230, 229, 235, 273ltsub23d 11816 . . . . . . . . . . . . . . 15 (((((((𝐴 βŠ† ℝ ∧ (vol*β€˜π΄) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑒 ∈ ℝ ∧ 𝑒 < (vol*β€˜π΄))) ∧ (𝐴 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ≀ ((vol*β€˜π΄) + 1))) ∧ ((βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) βŠ† βˆͺ ran ((,) ∘ 𝑔) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑔)) ≀ ((vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)) + (((vol*β€˜π΄) βˆ’ 𝑒) / 2)))) ∧ (𝑠 ∈ (Clsdβ€˜(topGenβ€˜ran (,))) ∧ (𝑠 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ ((vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) βˆ’ (((vol*β€˜π΄) βˆ’ 𝑒) / 2)) < (vol*β€˜π‘ )))) β†’ ((vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) βˆ’ (vol*β€˜π‘ )) < (((vol*β€˜π΄) βˆ’ 𝑒) / 2))
275214, 236, 229, 272, 274lelttrd 11369 . . . . . . . . . . . . . 14 (((((((𝐴 βŠ† ℝ ∧ (vol*β€˜π΄) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑒 ∈ ℝ ∧ 𝑒 < (vol*β€˜π΄))) ∧ (𝐴 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ≀ ((vol*β€˜π΄) + 1))) ∧ ((βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) βŠ† βˆͺ ran ((,) ∘ 𝑔) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑔)) ≀ ((vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)) + (((vol*β€˜π΄) βˆ’ 𝑒) / 2)))) ∧ (𝑠 ∈ (Clsdβ€˜(topGenβ€˜ran (,))) ∧ (𝑠 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ ((vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) βˆ’ (((vol*β€˜π΄) βˆ’ 𝑒) / 2)) < (vol*β€˜π‘ )))) β†’ (vol*β€˜(𝐴 βˆ– 𝑠)) < (((vol*β€˜π΄) βˆ’ 𝑒) / 2))
276216ad4antr 731 . . . . . . . . . . . . . . . 16 ((((((𝐴 βŠ† ℝ ∧ (vol*β€˜π΄) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑒 ∈ ℝ ∧ 𝑒 < (vol*β€˜π΄))) ∧ (𝐴 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ≀ ((vol*β€˜π΄) + 1))) ∧ ((βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) βŠ† βˆͺ ran ((,) ∘ 𝑔) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑔)) ≀ ((vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)) + (((vol*β€˜π΄) βˆ’ 𝑒) / 2)))) β†’ (vol*β€˜(𝐴 ∩ βˆͺ ran ((,) ∘ 𝑔))) ∈ ℝ)
277126, 132jctil 521 . . . . . . . . . . . . . . . . . 18 (((((𝐴 βŠ† ℝ ∧ (vol*β€˜π΄) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑒 ∈ ℝ ∧ 𝑒 < (vol*β€˜π΄))) ∧ (𝐴 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ≀ ((vol*β€˜π΄) + 1))) β†’ ((vol*β€˜βˆͺ ran ((,) ∘ 𝑔)) ∈ ℝ* ∧ ((vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)) + (((vol*β€˜π΄) βˆ’ 𝑒) / 2)) ∈ ℝ))
278 simpr 486 . . . . . . . . . . . . . . . . . . 19 (((βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) βŠ† βˆͺ ran ((,) ∘ 𝑔) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑔)) ≀ ((vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)) + (((vol*β€˜π΄) βˆ’ 𝑒) / 2))) β†’ (vol*β€˜βˆͺ ran ((,) ∘ 𝑔)) ≀ ((vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)) + (((vol*β€˜π΄) βˆ’ 𝑒) / 2)))
279 ovolge0 24990 . . . . . . . . . . . . . . . . . . . 20 (βˆͺ ran ((,) ∘ 𝑔) βŠ† ℝ β†’ 0 ≀ (vol*β€˜βˆͺ ran ((,) ∘ 𝑔)))
280130, 279ax-mp 5 . . . . . . . . . . . . . . . . . . 19 0 ≀ (vol*β€˜βˆͺ ran ((,) ∘ 𝑔))
281278, 280jctil 521 . . . . . . . . . . . . . . . . . 18 (((βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) βŠ† βˆͺ ran ((,) ∘ 𝑔) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑔)) ≀ ((vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)) + (((vol*β€˜π΄) βˆ’ 𝑒) / 2))) β†’ (0 ≀ (vol*β€˜βˆͺ ran ((,) ∘ 𝑔)) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑔)) ≀ ((vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)) + (((vol*β€˜π΄) βˆ’ 𝑒) / 2))))
282 xrrege0 13150 . . . . . . . . . . . . . . . . . 18 ((((vol*β€˜βˆͺ ran ((,) ∘ 𝑔)) ∈ ℝ* ∧ ((vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)) + (((vol*β€˜π΄) βˆ’ 𝑒) / 2)) ∈ ℝ) ∧ (0 ≀ (vol*β€˜βˆͺ ran ((,) ∘ 𝑔)) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑔)) ≀ ((vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)) + (((vol*β€˜π΄) βˆ’ 𝑒) / 2)))) β†’ (vol*β€˜βˆͺ ran ((,) ∘ 𝑔)) ∈ ℝ)
283277, 281, 282syl2an 597 . . . . . . . . . . . . . . . . 17 ((((((𝐴 βŠ† ℝ ∧ (vol*β€˜π΄) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑒 ∈ ℝ ∧ 𝑒 < (vol*β€˜π΄))) ∧ (𝐴 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ≀ ((vol*β€˜π΄) + 1))) ∧ ((βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) βŠ† βˆͺ ran ((,) ∘ 𝑔) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑔)) ≀ ((vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)) + (((vol*β€˜π΄) βˆ’ 𝑒) / 2)))) β†’ (vol*β€˜βˆͺ ran ((,) ∘ 𝑔)) ∈ ℝ)
284 difss 4131 . . . . . . . . . . . . . . . . . 18 (βˆͺ ran ((,) ∘ 𝑔) βˆ– (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)) βŠ† βˆͺ ran ((,) ∘ 𝑔)
285 ovolsscl 24995 . . . . . . . . . . . . . . . . . 18 (((βˆͺ ran ((,) ∘ 𝑔) βˆ– (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)) βŠ† βˆͺ ran ((,) ∘ 𝑔) ∧ βˆͺ ran ((,) ∘ 𝑔) βŠ† ℝ ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑔)) ∈ ℝ) β†’ (vol*β€˜(βˆͺ ran ((,) ∘ 𝑔) βˆ– (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴))) ∈ ℝ)
286284, 130, 285mp3an12 1452 . . . . . . . . . . . . . . . . 17 ((vol*β€˜βˆͺ ran ((,) ∘ 𝑔)) ∈ ℝ β†’ (vol*β€˜(βˆͺ ran ((,) ∘ 𝑔) βˆ– (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴))) ∈ ℝ)
287283, 286syl 17 . . . . . . . . . . . . . . . 16 ((((((𝐴 βŠ† ℝ ∧ (vol*β€˜π΄) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑒 ∈ ℝ ∧ 𝑒 < (vol*β€˜π΄))) ∧ (𝐴 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ≀ ((vol*β€˜π΄) + 1))) ∧ ((βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) βŠ† βˆͺ ran ((,) ∘ 𝑔) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑔)) ≀ ((vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)) + (((vol*β€˜π΄) βˆ’ 𝑒) / 2)))) β†’ (vol*β€˜(βˆͺ ran ((,) ∘ 𝑔) βˆ– (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴))) ∈ ℝ)
288 ssun2 4173 . . . . . . . . . . . . . . . . . . 19 (βˆͺ ran ((,) ∘ 𝑔) ∩ 𝐴) βŠ† ((βˆͺ ran ((,) ∘ 𝑔) βˆ– βˆͺ ran ((,) ∘ 𝑓)) βˆͺ (βˆͺ ran ((,) ∘ 𝑔) ∩ 𝐴))
289 incom 4201 . . . . . . . . . . . . . . . . . . 19 (𝐴 ∩ βˆͺ ran ((,) ∘ 𝑔)) = (βˆͺ ran ((,) ∘ 𝑔) ∩ 𝐴)
290 difdif2 4286 . . . . . . . . . . . . . . . . . . 19 (βˆͺ ran ((,) ∘ 𝑔) βˆ– (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)) = ((βˆͺ ran ((,) ∘ 𝑔) βˆ– βˆͺ ran ((,) ∘ 𝑓)) βˆͺ (βˆͺ ran ((,) ∘ 𝑔) ∩ 𝐴))
291288, 289, 2903sstr4i 4025 . . . . . . . . . . . . . . . . . 18 (𝐴 ∩ βˆͺ ran ((,) ∘ 𝑔)) βŠ† (βˆͺ ran ((,) ∘ 𝑔) βˆ– (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴))
292284, 130sstri 3991 . . . . . . . . . . . . . . . . . 18 (βˆͺ ran ((,) ∘ 𝑔) βˆ– (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)) βŠ† ℝ
293291, 292pm3.2i 472 . . . . . . . . . . . . . . . . 17 ((𝐴 ∩ βˆͺ ran ((,) ∘ 𝑔)) βŠ† (βˆͺ ran ((,) ∘ 𝑔) βˆ– (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)) ∧ (βˆͺ ran ((,) ∘ 𝑔) βˆ– (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)) βŠ† ℝ)
294 ovolss 24994 . . . . . . . . . . . . . . . . 17 (((𝐴 ∩ βˆͺ ran ((,) ∘ 𝑔)) βŠ† (βˆͺ ran ((,) ∘ 𝑔) βˆ– (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)) ∧ (βˆͺ ran ((,) ∘ 𝑔) βˆ– (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)) βŠ† ℝ) β†’ (vol*β€˜(𝐴 ∩ βˆͺ ran ((,) ∘ 𝑔))) ≀ (vol*β€˜(βˆͺ ran ((,) ∘ 𝑔) βˆ– (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴))))
295293, 294mp1i 13 . . . . . . . . . . . . . . . 16 ((((((𝐴 βŠ† ℝ ∧ (vol*β€˜π΄) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑒 ∈ ℝ ∧ 𝑒 < (vol*β€˜π΄))) ∧ (𝐴 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ≀ ((vol*β€˜π΄) + 1))) ∧ ((βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) βŠ† βˆͺ ran ((,) ∘ 𝑔) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑔)) ≀ ((vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)) + (((vol*β€˜π΄) βˆ’ 𝑒) / 2)))) β†’ (vol*β€˜(𝐴 ∩ βˆͺ ran ((,) ∘ 𝑔))) ≀ (vol*β€˜(βˆͺ ran ((,) ∘ 𝑔) βˆ– (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴))))
296 opnmbl 25111 . . . . . . . . . . . . . . . . . . . . . . 23 (βˆͺ ran ((,) ∘ 𝑓) ∈ (topGenβ€˜ran (,)) β†’ βˆͺ ran ((,) ∘ 𝑓) ∈ dom vol)
297159, 296ax-mp 5 . . . . . . . . . . . . . . . . . . . . . 22 βˆͺ ran ((,) ∘ 𝑓) ∈ dom vol
298 difmbl 25052 . . . . . . . . . . . . . . . . . . . . . 22 ((βˆͺ ran ((,) ∘ 𝑓) ∈ dom vol ∧ 𝐴 ∈ dom vol) β†’ (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) ∈ dom vol)
299297, 298mpan 689 . . . . . . . . . . . . . . . . . . . . 21 (𝐴 ∈ dom vol β†’ (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) ∈ dom vol)
300299ad4antlr 732 . . . . . . . . . . . . . . . . . . . 20 ((((((𝐴 βŠ† ℝ ∧ (vol*β€˜π΄) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑒 ∈ ℝ ∧ 𝑒 < (vol*β€˜π΄))) ∧ (𝐴 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ≀ ((vol*β€˜π΄) + 1))) ∧ ((βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) βŠ† βˆͺ ran ((,) ∘ 𝑔) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑔)) ≀ ((vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)) + (((vol*β€˜π΄) βˆ’ 𝑒) / 2)))) β†’ (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) ∈ dom vol)
301 mblsplit 25041 . . . . . . . . . . . . . . . . . . . . 21 (((βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) ∈ dom vol ∧ βˆͺ ran ((,) ∘ 𝑔) βŠ† ℝ ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑔)) ∈ ℝ) β†’ (vol*β€˜βˆͺ ran ((,) ∘ 𝑔)) = ((vol*β€˜(βˆͺ ran ((,) ∘ 𝑔) ∩ (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴))) + (vol*β€˜(βˆͺ ran ((,) ∘ 𝑔) βˆ– (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)))))
302130, 301mp3an2 1450 . . . . . . . . . . . . . . . . . . . 20 (((βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) ∈ dom vol ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑔)) ∈ ℝ) β†’ (vol*β€˜βˆͺ ran ((,) ∘ 𝑔)) = ((vol*β€˜(βˆͺ ran ((,) ∘ 𝑔) ∩ (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴))) + (vol*β€˜(βˆͺ ran ((,) ∘ 𝑔) βˆ– (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)))))
303300, 283, 302syl2anc 585 . . . . . . . . . . . . . . . . . . 19 ((((((𝐴 βŠ† ℝ ∧ (vol*β€˜π΄) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑒 ∈ ℝ ∧ 𝑒 < (vol*β€˜π΄))) ∧ (𝐴 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ≀ ((vol*β€˜π΄) + 1))) ∧ ((βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) βŠ† βˆͺ ran ((,) ∘ 𝑔) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑔)) ≀ ((vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)) + (((vol*β€˜π΄) βˆ’ 𝑒) / 2)))) β†’ (vol*β€˜βˆͺ ran ((,) ∘ 𝑔)) = ((vol*β€˜(βˆͺ ran ((,) ∘ 𝑔) ∩ (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴))) + (vol*β€˜(βˆͺ ran ((,) ∘ 𝑔) βˆ– (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)))))
304 sseqin2 4215 . . . . . . . . . . . . . . . . . . . . . . 23 ((βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) βŠ† βˆͺ ran ((,) ∘ 𝑔) ↔ (βˆͺ ran ((,) ∘ 𝑔) ∩ (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)) = (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴))
305304biimpi 215 . . . . . . . . . . . . . . . . . . . . . 22 ((βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) βŠ† βˆͺ ran ((,) ∘ 𝑔) β†’ (βˆͺ ran ((,) ∘ 𝑔) ∩ (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)) = (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴))
306305fveq2d 6893 . . . . . . . . . . . . . . . . . . . . 21 ((βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) βŠ† βˆͺ ran ((,) ∘ 𝑔) β†’ (vol*β€˜(βˆͺ ran ((,) ∘ 𝑔) ∩ (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴))) = (vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)))
307306oveq1d 7421 . . . . . . . . . . . . . . . . . . . 20 ((βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) βŠ† βˆͺ ran ((,) ∘ 𝑔) β†’ ((vol*β€˜(βˆͺ ran ((,) ∘ 𝑔) ∩ (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴))) + (vol*β€˜(βˆͺ ran ((,) ∘ 𝑔) βˆ– (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)))) = ((vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)) + (vol*β€˜(βˆͺ ran ((,) ∘ 𝑔) βˆ– (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)))))
308307ad2antrl 727 . . . . . . . . . . . . . . . . . . 19 ((((((𝐴 βŠ† ℝ ∧ (vol*β€˜π΄) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑒 ∈ ℝ ∧ 𝑒 < (vol*β€˜π΄))) ∧ (𝐴 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ≀ ((vol*β€˜π΄) + 1))) ∧ ((βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) βŠ† βˆͺ ran ((,) ∘ 𝑔) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑔)) ≀ ((vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)) + (((vol*β€˜π΄) βˆ’ 𝑒) / 2)))) β†’ ((vol*β€˜(βˆͺ ran ((,) ∘ 𝑔) ∩ (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴))) + (vol*β€˜(βˆͺ ran ((,) ∘ 𝑔) βˆ– (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)))) = ((vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)) + (vol*β€˜(βˆͺ ran ((,) ∘ 𝑔) βˆ– (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)))))
309303, 308eqtr2d 2774 . . . . . . . . . . . . . . . . . 18 ((((((𝐴 βŠ† ℝ ∧ (vol*β€˜π΄) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑒 ∈ ℝ ∧ 𝑒 < (vol*β€˜π΄))) ∧ (𝐴 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ≀ ((vol*β€˜π΄) + 1))) ∧ ((βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) βŠ† βˆͺ ran ((,) ∘ 𝑔) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑔)) ≀ ((vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)) + (((vol*β€˜π΄) βˆ’ 𝑒) / 2)))) β†’ ((vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)) + (vol*β€˜(βˆͺ ran ((,) ∘ 𝑔) βˆ– (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)))) = (vol*β€˜βˆͺ ran ((,) ∘ 𝑔)))
310283recnd 11239 . . . . . . . . . . . . . . . . . . 19 ((((((𝐴 βŠ† ℝ ∧ (vol*β€˜π΄) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑒 ∈ ℝ ∧ 𝑒 < (vol*β€˜π΄))) ∧ (𝐴 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ≀ ((vol*β€˜π΄) + 1))) ∧ ((βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) βŠ† βˆͺ ran ((,) ∘ 𝑔) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑔)) ≀ ((vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)) + (((vol*β€˜π΄) βˆ’ 𝑒) / 2)))) β†’ (vol*β€˜βˆͺ ran ((,) ∘ 𝑔)) ∈ β„‚)
31197adantr 482 . . . . . . . . . . . . . . . . . . . 20 ((((((𝐴 βŠ† ℝ ∧ (vol*β€˜π΄) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑒 ∈ ℝ ∧ 𝑒 < (vol*β€˜π΄))) ∧ (𝐴 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ≀ ((vol*β€˜π΄) + 1))) ∧ ((βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) βŠ† βˆͺ ran ((,) ∘ 𝑔) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑔)) ≀ ((vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)) + (((vol*β€˜π΄) βˆ’ 𝑒) / 2)))) β†’ (vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)) ∈ ℝ)
312311recnd 11239 . . . . . . . . . . . . . . . . . . 19 ((((((𝐴 βŠ† ℝ ∧ (vol*β€˜π΄) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑒 ∈ ℝ ∧ 𝑒 < (vol*β€˜π΄))) ∧ (𝐴 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ≀ ((vol*β€˜π΄) + 1))) ∧ ((βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) βŠ† βˆͺ ran ((,) ∘ 𝑔) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑔)) ≀ ((vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)) + (((vol*β€˜π΄) βˆ’ 𝑒) / 2)))) β†’ (vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)) ∈ β„‚)
313287recnd 11239 . . . . . . . . . . . . . . . . . . 19 ((((((𝐴 βŠ† ℝ ∧ (vol*β€˜π΄) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑒 ∈ ℝ ∧ 𝑒 < (vol*β€˜π΄))) ∧ (𝐴 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ≀ ((vol*β€˜π΄) + 1))) ∧ ((βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) βŠ† βˆͺ ran ((,) ∘ 𝑔) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑔)) ≀ ((vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)) + (((vol*β€˜π΄) βˆ’ 𝑒) / 2)))) β†’ (vol*β€˜(βˆͺ ran ((,) ∘ 𝑔) βˆ– (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴))) ∈ β„‚)
314310, 312, 313subaddd 11586 . . . . . . . . . . . . . . . . . 18 ((((((𝐴 βŠ† ℝ ∧ (vol*β€˜π΄) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑒 ∈ ℝ ∧ 𝑒 < (vol*β€˜π΄))) ∧ (𝐴 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ≀ ((vol*β€˜π΄) + 1))) ∧ ((βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) βŠ† βˆͺ ran ((,) ∘ 𝑔) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑔)) ≀ ((vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)) + (((vol*β€˜π΄) βˆ’ 𝑒) / 2)))) β†’ (((vol*β€˜βˆͺ ran ((,) ∘ 𝑔)) βˆ’ (vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴))) = (vol*β€˜(βˆͺ ran ((,) ∘ 𝑔) βˆ– (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴))) ↔ ((vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)) + (vol*β€˜(βˆͺ ran ((,) ∘ 𝑔) βˆ– (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)))) = (vol*β€˜βˆͺ ran ((,) ∘ 𝑔))))
315309, 314mpbird 257 . . . . . . . . . . . . . . . . 17 ((((((𝐴 βŠ† ℝ ∧ (vol*β€˜π΄) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑒 ∈ ℝ ∧ 𝑒 < (vol*β€˜π΄))) ∧ (𝐴 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ≀ ((vol*β€˜π΄) + 1))) ∧ ((βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) βŠ† βˆͺ ran ((,) ∘ 𝑔) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑔)) ≀ ((vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)) + (((vol*β€˜π΄) βˆ’ 𝑒) / 2)))) β†’ ((vol*β€˜βˆͺ ran ((,) ∘ 𝑔)) βˆ’ (vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴))) = (vol*β€˜(βˆͺ ran ((,) ∘ 𝑔) βˆ– (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴))))
316 simprr 772 . . . . . . . . . . . . . . . . . 18 ((((((𝐴 βŠ† ℝ ∧ (vol*β€˜π΄) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑒 ∈ ℝ ∧ 𝑒 < (vol*β€˜π΄))) ∧ (𝐴 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ≀ ((vol*β€˜π΄) + 1))) ∧ ((βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) βŠ† βˆͺ ran ((,) ∘ 𝑔) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑔)) ≀ ((vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)) + (((vol*β€˜π΄) βˆ’ 𝑒) / 2)))) β†’ (vol*β€˜βˆͺ ran ((,) ∘ 𝑔)) ≀ ((vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)) + (((vol*β€˜π΄) βˆ’ 𝑒) / 2)))
317283, 311, 228lesubadd2d 11810 . . . . . . . . . . . . . . . . . 18 ((((((𝐴 βŠ† ℝ ∧ (vol*β€˜π΄) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑒 ∈ ℝ ∧ 𝑒 < (vol*β€˜π΄))) ∧ (𝐴 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ≀ ((vol*β€˜π΄) + 1))) ∧ ((βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) βŠ† βˆͺ ran ((,) ∘ 𝑔) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑔)) ≀ ((vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)) + (((vol*β€˜π΄) βˆ’ 𝑒) / 2)))) β†’ (((vol*β€˜βˆͺ ran ((,) ∘ 𝑔)) βˆ’ (vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴))) ≀ (((vol*β€˜π΄) βˆ’ 𝑒) / 2) ↔ (vol*β€˜βˆͺ ran ((,) ∘ 𝑔)) ≀ ((vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)) + (((vol*β€˜π΄) βˆ’ 𝑒) / 2))))
318316, 317mpbird 257 . . . . . . . . . . . . . . . . 17 ((((((𝐴 βŠ† ℝ ∧ (vol*β€˜π΄) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑒 ∈ ℝ ∧ 𝑒 < (vol*β€˜π΄))) ∧ (𝐴 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ≀ ((vol*β€˜π΄) + 1))) ∧ ((βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) βŠ† βˆͺ ran ((,) ∘ 𝑔) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑔)) ≀ ((vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)) + (((vol*β€˜π΄) βˆ’ 𝑒) / 2)))) β†’ ((vol*β€˜βˆͺ ran ((,) ∘ 𝑔)) βˆ’ (vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴))) ≀ (((vol*β€˜π΄) βˆ’ 𝑒) / 2))
319315, 318eqbrtrrd 5172 . . . . . . . . . . . . . . . 16 ((((((𝐴 βŠ† ℝ ∧ (vol*β€˜π΄) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑒 ∈ ℝ ∧ 𝑒 < (vol*β€˜π΄))) ∧ (𝐴 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ≀ ((vol*β€˜π΄) + 1))) ∧ ((βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) βŠ† βˆͺ ran ((,) ∘ 𝑔) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑔)) ≀ ((vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)) + (((vol*β€˜π΄) βˆ’ 𝑒) / 2)))) β†’ (vol*β€˜(βˆͺ ran ((,) ∘ 𝑔) βˆ– (βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴))) ≀ (((vol*β€˜π΄) βˆ’ 𝑒) / 2))
320276, 287, 228, 295, 319letrd 11368 . . . . . . . . . . . . . . 15 ((((((𝐴 βŠ† ℝ ∧ (vol*β€˜π΄) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑒 ∈ ℝ ∧ 𝑒 < (vol*β€˜π΄))) ∧ (𝐴 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ≀ ((vol*β€˜π΄) + 1))) ∧ ((βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) βŠ† βˆͺ ran ((,) ∘ 𝑔) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑔)) ≀ ((vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)) + (((vol*β€˜π΄) βˆ’ 𝑒) / 2)))) β†’ (vol*β€˜(𝐴 ∩ βˆͺ ran ((,) ∘ 𝑔))) ≀ (((vol*β€˜π΄) βˆ’ 𝑒) / 2))
321320adantr 482 . . . . . . . . . . . . . 14 (((((((𝐴 βŠ† ℝ ∧ (vol*β€˜π΄) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑒 ∈ ℝ ∧ 𝑒 < (vol*β€˜π΄))) ∧ (𝐴 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ≀ ((vol*β€˜π΄) + 1))) ∧ ((βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) βŠ† βˆͺ ran ((,) ∘ 𝑔) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑔)) ≀ ((vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)) + (((vol*β€˜π΄) βˆ’ 𝑒) / 2)))) ∧ (𝑠 ∈ (Clsdβ€˜(topGenβ€˜ran (,))) ∧ (𝑠 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ ((vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) βˆ’ (((vol*β€˜π΄) βˆ’ 𝑒) / 2)) < (vol*β€˜π‘ )))) β†’ (vol*β€˜(𝐴 ∩ βˆͺ ran ((,) ∘ 𝑔))) ≀ (((vol*β€˜π΄) βˆ’ 𝑒) / 2))
322214, 217, 229, 229, 275, 321ltleaddd 11832 . . . . . . . . . . . . 13 (((((((𝐴 βŠ† ℝ ∧ (vol*β€˜π΄) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑒 ∈ ℝ ∧ 𝑒 < (vol*β€˜π΄))) ∧ (𝐴 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ≀ ((vol*β€˜π΄) + 1))) ∧ ((βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) βŠ† βˆͺ ran ((,) ∘ 𝑔) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑔)) ≀ ((vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)) + (((vol*β€˜π΄) βˆ’ 𝑒) / 2)))) ∧ (𝑠 ∈ (Clsdβ€˜(topGenβ€˜ran (,))) ∧ (𝑠 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ ((vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) βˆ’ (((vol*β€˜π΄) βˆ’ 𝑒) / 2)) < (vol*β€˜π‘ )))) β†’ ((vol*β€˜(𝐴 βˆ– 𝑠)) + (vol*β€˜(𝐴 ∩ βˆͺ ran ((,) ∘ 𝑔)))) < ((((vol*β€˜π΄) βˆ’ 𝑒) / 2) + (((vol*β€˜π΄) βˆ’ 𝑒) / 2)))
32398recnd 11239 . . . . . . . . . . . . . . . . 17 (((vol*β€˜π΄) ∈ ℝ ∧ 𝑒 ∈ ℝ) β†’ ((vol*β€˜π΄) βˆ’ 𝑒) ∈ β„‚)
3243232halvesd 12455 . . . . . . . . . . . . . . . 16 (((vol*β€˜π΄) ∈ ℝ ∧ 𝑒 ∈ ℝ) β†’ ((((vol*β€˜π΄) βˆ’ 𝑒) / 2) + (((vol*β€˜π΄) βˆ’ 𝑒) / 2)) = ((vol*β€˜π΄) βˆ’ 𝑒))
325324adantll 713 . . . . . . . . . . . . . . 15 (((𝐴 βŠ† ℝ ∧ (vol*β€˜π΄) ∈ ℝ) ∧ 𝑒 ∈ ℝ) β†’ ((((vol*β€˜π΄) βˆ’ 𝑒) / 2) + (((vol*β€˜π΄) βˆ’ 𝑒) / 2)) = ((vol*β€˜π΄) βˆ’ 𝑒))
326325ad2ant2r 746 . . . . . . . . . . . . . 14 ((((𝐴 βŠ† ℝ ∧ (vol*β€˜π΄) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑒 ∈ ℝ ∧ 𝑒 < (vol*β€˜π΄))) β†’ ((((vol*β€˜π΄) βˆ’ 𝑒) / 2) + (((vol*β€˜π΄) βˆ’ 𝑒) / 2)) = ((vol*β€˜π΄) βˆ’ 𝑒))
327326ad3antrrr 729 . . . . . . . . . . . . 13 (((((((𝐴 βŠ† ℝ ∧ (vol*β€˜π΄) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑒 ∈ ℝ ∧ 𝑒 < (vol*β€˜π΄))) ∧ (𝐴 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ≀ ((vol*β€˜π΄) + 1))) ∧ ((βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) βŠ† βˆͺ ran ((,) ∘ 𝑔) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑔)) ≀ ((vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)) + (((vol*β€˜π΄) βˆ’ 𝑒) / 2)))) ∧ (𝑠 ∈ (Clsdβ€˜(topGenβ€˜ran (,))) ∧ (𝑠 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ ((vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) βˆ’ (((vol*β€˜π΄) βˆ’ 𝑒) / 2)) < (vol*β€˜π‘ )))) β†’ ((((vol*β€˜π΄) βˆ’ 𝑒) / 2) + (((vol*β€˜π΄) βˆ’ 𝑒) / 2)) = ((vol*β€˜π΄) βˆ’ 𝑒))
328322, 327breqtrd 5174 . . . . . . . . . . . 12 (((((((𝐴 βŠ† ℝ ∧ (vol*β€˜π΄) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑒 ∈ ℝ ∧ 𝑒 < (vol*β€˜π΄))) ∧ (𝐴 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ≀ ((vol*β€˜π΄) + 1))) ∧ ((βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) βŠ† βˆͺ ran ((,) ∘ 𝑔) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑔)) ≀ ((vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)) + (((vol*β€˜π΄) βˆ’ 𝑒) / 2)))) ∧ (𝑠 ∈ (Clsdβ€˜(topGenβ€˜ran (,))) ∧ (𝑠 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ ((vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) βˆ’ (((vol*β€˜π΄) βˆ’ 𝑒) / 2)) < (vol*β€˜π‘ )))) β†’ ((vol*β€˜(𝐴 βˆ– 𝑠)) + (vol*β€˜(𝐴 ∩ βˆͺ ran ((,) ∘ 𝑔)))) < ((vol*β€˜π΄) βˆ’ 𝑒))
329211, 218, 220, 227, 328lelttrd 11369 . . . . . . . . . . 11 (((((((𝐴 βŠ† ℝ ∧ (vol*β€˜π΄) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑒 ∈ ℝ ∧ 𝑒 < (vol*β€˜π΄))) ∧ (𝐴 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ≀ ((vol*β€˜π΄) + 1))) ∧ ((βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) βŠ† βˆͺ ran ((,) ∘ 𝑔) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑔)) ≀ ((vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)) + (((vol*β€˜π΄) βˆ’ 𝑒) / 2)))) ∧ (𝑠 ∈ (Clsdβ€˜(topGenβ€˜ran (,))) ∧ (𝑠 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ ((vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) βˆ’ (((vol*β€˜π΄) βˆ’ 𝑒) / 2)) < (vol*β€˜π‘ )))) β†’ (vol*β€˜((𝐴 βˆ– 𝑠) βˆͺ (𝐴 ∩ βˆͺ ran ((,) ∘ 𝑔)))) < ((vol*β€˜π΄) βˆ’ 𝑒))
330205, 329eqbrtrid 5183 . . . . . . . . . 10 (((((((𝐴 βŠ† ℝ ∧ (vol*β€˜π΄) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑒 ∈ ℝ ∧ 𝑒 < (vol*β€˜π΄))) ∧ (𝐴 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ≀ ((vol*β€˜π΄) + 1))) ∧ ((βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) βŠ† βˆͺ ran ((,) ∘ 𝑔) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑔)) ≀ ((vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)) + (((vol*β€˜π΄) βˆ’ 𝑒) / 2)))) ∧ (𝑠 ∈ (Clsdβ€˜(topGenβ€˜ran (,))) ∧ (𝑠 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ ((vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) βˆ’ (((vol*β€˜π΄) βˆ’ 𝑒) / 2)) < (vol*β€˜π‘ )))) β†’ (vol*β€˜(𝐴 βˆ– (𝑠 βˆ– βˆͺ ran ((,) ∘ 𝑔)))) < ((vol*β€˜π΄) βˆ’ 𝑒))
331200, 201, 203, 330ltsub13d 11817 . . . . . . . . 9 (((((((𝐴 βŠ† ℝ ∧ (vol*β€˜π΄) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑒 ∈ ℝ ∧ 𝑒 < (vol*β€˜π΄))) ∧ (𝐴 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ≀ ((vol*β€˜π΄) + 1))) ∧ ((βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) βŠ† βˆͺ ran ((,) ∘ 𝑔) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑔)) ≀ ((vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)) + (((vol*β€˜π΄) βˆ’ 𝑒) / 2)))) ∧ (𝑠 ∈ (Clsdβ€˜(topGenβ€˜ran (,))) ∧ (𝑠 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ ((vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) βˆ’ (((vol*β€˜π΄) βˆ’ 𝑒) / 2)) < (vol*β€˜π‘ )))) β†’ 𝑒 < ((vol*β€˜π΄) βˆ’ (vol*β€˜(𝐴 βˆ– (𝑠 βˆ– βˆͺ ran ((,) ∘ 𝑔))))))
332 opnmbl 25111 . . . . . . . . . . . . . . . 16 (βˆͺ ran ((,) ∘ 𝑔) ∈ (topGenβ€˜ran (,)) β†’ βˆͺ ran ((,) ∘ 𝑔) ∈ dom vol)
333172, 332ax-mp 5 . . . . . . . . . . . . . . 15 βˆͺ ran ((,) ∘ 𝑔) ∈ dom vol
334 difmbl 25052 . . . . . . . . . . . . . . 15 ((𝑠 ∈ dom vol ∧ βˆͺ ran ((,) ∘ 𝑔) ∈ dom vol) β†’ (𝑠 βˆ– βˆͺ ran ((,) ∘ 𝑔)) ∈ dom vol)
335245, 333, 334sylancl 587 . . . . . . . . . . . . . 14 (𝑠 ∈ (Clsdβ€˜(topGenβ€˜ran (,))) β†’ (𝑠 βˆ– βˆͺ ran ((,) ∘ 𝑔)) ∈ dom vol)
336 mblvol 25039 . . . . . . . . . . . . . 14 ((𝑠 βˆ– βˆͺ ran ((,) ∘ 𝑔)) ∈ dom vol β†’ (volβ€˜(𝑠 βˆ– βˆͺ ran ((,) ∘ 𝑔))) = (vol*β€˜(𝑠 βˆ– βˆͺ ran ((,) ∘ 𝑔))))
337335, 336syl 17 . . . . . . . . . . . . 13 (𝑠 ∈ (Clsdβ€˜(topGenβ€˜ran (,))) β†’ (volβ€˜(𝑠 βˆ– βˆͺ ran ((,) ∘ 𝑔))) = (vol*β€˜(𝑠 βˆ– βˆͺ ran ((,) ∘ 𝑔))))
338337ad2antrl 727 . . . . . . . . . . . 12 ((((βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) βŠ† βˆͺ ran ((,) ∘ 𝑔) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑔)) ≀ ((vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)) + (((vol*β€˜π΄) βˆ’ 𝑒) / 2))) ∧ (𝑠 ∈ (Clsdβ€˜(topGenβ€˜ran (,))) ∧ (𝑠 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ ((vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) βˆ’ (((vol*β€˜π΄) βˆ’ 𝑒) / 2)) < (vol*β€˜π‘ )))) β†’ (volβ€˜(𝑠 βˆ– βˆͺ ran ((,) ∘ 𝑔))) = (vol*β€˜(𝑠 βˆ– βˆͺ ran ((,) ∘ 𝑔))))
339 sseqin2 4215 . . . . . . . . . . . . . . . 16 ((𝑠 βˆ– βˆͺ ran ((,) ∘ 𝑔)) βŠ† 𝐴 ↔ (𝐴 ∩ (𝑠 βˆ– βˆͺ ran ((,) ∘ 𝑔))) = (𝑠 βˆ– βˆͺ ran ((,) ∘ 𝑔)))
340184, 339sylib 217 . . . . . . . . . . . . . . 15 (((βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) βŠ† βˆͺ ran ((,) ∘ 𝑔) ∧ 𝑠 βŠ† βˆͺ ran ((,) ∘ 𝑓)) β†’ (𝐴 ∩ (𝑠 βˆ– βˆͺ ran ((,) ∘ 𝑔))) = (𝑠 βˆ– βˆͺ ran ((,) ∘ 𝑔)))
341340fveq2d 6893 . . . . . . . . . . . . . 14 (((βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) βŠ† βˆͺ ran ((,) ∘ 𝑔) ∧ 𝑠 βŠ† βˆͺ ran ((,) ∘ 𝑓)) β†’ (vol*β€˜(𝐴 ∩ (𝑠 βˆ– βˆͺ ran ((,) ∘ 𝑔)))) = (vol*β€˜(𝑠 βˆ– βˆͺ ran ((,) ∘ 𝑔))))
342341adantrr 716 . . . . . . . . . . . . 13 (((βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) βŠ† βˆͺ ran ((,) ∘ 𝑔) ∧ (𝑠 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ ((vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) βˆ’ (((vol*β€˜π΄) βˆ’ 𝑒) / 2)) < (vol*β€˜π‘ ))) β†’ (vol*β€˜(𝐴 ∩ (𝑠 βˆ– βˆͺ ran ((,) ∘ 𝑔)))) = (vol*β€˜(𝑠 βˆ– βˆͺ ran ((,) ∘ 𝑔))))
343342ad2ant2rl 748 . . . . . . . . . . . 12 ((((βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) βŠ† βˆͺ ran ((,) ∘ 𝑔) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑔)) ≀ ((vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)) + (((vol*β€˜π΄) βˆ’ 𝑒) / 2))) ∧ (𝑠 ∈ (Clsdβ€˜(topGenβ€˜ran (,))) ∧ (𝑠 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ ((vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) βˆ’ (((vol*β€˜π΄) βˆ’ 𝑒) / 2)) < (vol*β€˜π‘ )))) β†’ (vol*β€˜(𝐴 ∩ (𝑠 βˆ– βˆͺ ran ((,) ∘ 𝑔)))) = (vol*β€˜(𝑠 βˆ– βˆͺ ran ((,) ∘ 𝑔))))
344338, 343eqtr4d 2776 . . . . . . . . . . 11 ((((βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) βŠ† βˆͺ ran ((,) ∘ 𝑔) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑔)) ≀ ((vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)) + (((vol*β€˜π΄) βˆ’ 𝑒) / 2))) ∧ (𝑠 ∈ (Clsdβ€˜(topGenβ€˜ran (,))) ∧ (𝑠 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ ((vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) βˆ’ (((vol*β€˜π΄) βˆ’ 𝑒) / 2)) < (vol*β€˜π‘ )))) β†’ (volβ€˜(𝑠 βˆ– βˆͺ ran ((,) ∘ 𝑔))) = (vol*β€˜(𝐴 ∩ (𝑠 βˆ– βˆͺ ran ((,) ∘ 𝑔)))))
345344adantll 713 . . . . . . . . . 10 (((((((𝐴 βŠ† ℝ ∧ (vol*β€˜π΄) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑒 ∈ ℝ ∧ 𝑒 < (vol*β€˜π΄))) ∧ (𝐴 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ≀ ((vol*β€˜π΄) + 1))) ∧ ((βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) βŠ† βˆͺ ran ((,) ∘ 𝑔) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑔)) ≀ ((vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)) + (((vol*β€˜π΄) βˆ’ 𝑒) / 2)))) ∧ (𝑠 ∈ (Clsdβ€˜(topGenβ€˜ran (,))) ∧ (𝑠 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ ((vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) βˆ’ (((vol*β€˜π΄) βˆ’ 𝑒) / 2)) < (vol*β€˜π‘ )))) β†’ (volβ€˜(𝑠 βˆ– βˆͺ ran ((,) ∘ 𝑔))) = (vol*β€˜(𝐴 ∩ (𝑠 βˆ– βˆͺ ran ((,) ∘ 𝑔)))))
346335adantr 482 . . . . . . . . . . . 12 ((𝑠 ∈ (Clsdβ€˜(topGenβ€˜ran (,))) ∧ (𝑠 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ ((vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) βˆ’ (((vol*β€˜π΄) βˆ’ 𝑒) / 2)) < (vol*β€˜π‘ ))) β†’ (𝑠 βˆ– βˆͺ ran ((,) ∘ 𝑔)) ∈ dom vol)
347 simp-4l 782 . . . . . . . . . . . 12 ((((((𝐴 βŠ† ℝ ∧ (vol*β€˜π΄) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑒 ∈ ℝ ∧ 𝑒 < (vol*β€˜π΄))) ∧ (𝐴 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ≀ ((vol*β€˜π΄) + 1))) ∧ ((βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) βŠ† βˆͺ ran ((,) ∘ 𝑔) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑔)) ≀ ((vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)) + (((vol*β€˜π΄) βˆ’ 𝑒) / 2)))) β†’ (𝐴 βŠ† ℝ ∧ (vol*β€˜π΄) ∈ ℝ))
348 mblsplit 25041 . . . . . . . . . . . . . 14 (((𝑠 βˆ– βˆͺ ran ((,) ∘ 𝑔)) ∈ dom vol ∧ 𝐴 βŠ† ℝ ∧ (vol*β€˜π΄) ∈ ℝ) β†’ (vol*β€˜π΄) = ((vol*β€˜(𝐴 ∩ (𝑠 βˆ– βˆͺ ran ((,) ∘ 𝑔)))) + (vol*β€˜(𝐴 βˆ– (𝑠 βˆ– βˆͺ ran ((,) ∘ 𝑔))))))
3493483expb 1121 . . . . . . . . . . . . 13 (((𝑠 βˆ– βˆͺ ran ((,) ∘ 𝑔)) ∈ dom vol ∧ (𝐴 βŠ† ℝ ∧ (vol*β€˜π΄) ∈ ℝ)) β†’ (vol*β€˜π΄) = ((vol*β€˜(𝐴 ∩ (𝑠 βˆ– βˆͺ ran ((,) ∘ 𝑔)))) + (vol*β€˜(𝐴 βˆ– (𝑠 βˆ– βˆͺ ran ((,) ∘ 𝑔))))))
350349eqcomd 2739 . . . . . . . . . . . 12 (((𝑠 βˆ– βˆͺ ran ((,) ∘ 𝑔)) ∈ dom vol ∧ (𝐴 βŠ† ℝ ∧ (vol*β€˜π΄) ∈ ℝ)) β†’ ((vol*β€˜(𝐴 ∩ (𝑠 βˆ– βˆͺ ran ((,) ∘ 𝑔)))) + (vol*β€˜(𝐴 βˆ– (𝑠 βˆ– βˆͺ ran ((,) ∘ 𝑔))))) = (vol*β€˜π΄))
351346, 347, 350syl2anr 598 . . . . . . . . . . 11 (((((((𝐴 βŠ† ℝ ∧ (vol*β€˜π΄) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑒 ∈ ℝ ∧ 𝑒 < (vol*β€˜π΄))) ∧ (𝐴 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ≀ ((vol*β€˜π΄) + 1))) ∧ ((βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) βŠ† βˆͺ ran ((,) ∘ 𝑔) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑔)) ≀ ((vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)) + (((vol*β€˜π΄) βˆ’ 𝑒) / 2)))) ∧ (𝑠 ∈ (Clsdβ€˜(topGenβ€˜ran (,))) ∧ (𝑠 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ ((vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) βˆ’ (((vol*β€˜π΄) βˆ’ 𝑒) / 2)) < (vol*β€˜π‘ )))) β†’ ((vol*β€˜(𝐴 ∩ (𝑠 βˆ– βˆͺ ran ((,) ∘ 𝑔)))) + (vol*β€˜(𝐴 βˆ– (𝑠 βˆ– βˆͺ ran ((,) ∘ 𝑔))))) = (vol*β€˜π΄))
352 recn 11197 . . . . . . . . . . . . . 14 ((vol*β€˜π΄) ∈ ℝ β†’ (vol*β€˜π΄) ∈ β„‚)
353352adantl 483 . . . . . . . . . . . . 13 ((𝐴 βŠ† ℝ ∧ (vol*β€˜π΄) ∈ ℝ) β†’ (vol*β€˜π΄) ∈ β„‚)
354199recnd 11239 . . . . . . . . . . . . 13 ((𝐴 βŠ† ℝ ∧ (vol*β€˜π΄) ∈ ℝ) β†’ (vol*β€˜(𝐴 βˆ– (𝑠 βˆ– βˆͺ ran ((,) ∘ 𝑔)))) ∈ β„‚)
355 inss1 4228 . . . . . . . . . . . . . . 15 (𝐴 ∩ (𝑠 βˆ– βˆͺ ran ((,) ∘ 𝑔))) βŠ† 𝐴
356 ovolsscl 24995 . . . . . . . . . . . . . . 15 (((𝐴 ∩ (𝑠 βˆ– βˆͺ ran ((,) ∘ 𝑔))) βŠ† 𝐴 ∧ 𝐴 βŠ† ℝ ∧ (vol*β€˜π΄) ∈ ℝ) β†’ (vol*β€˜(𝐴 ∩ (𝑠 βˆ– βˆͺ ran ((,) ∘ 𝑔)))) ∈ ℝ)
357355, 356mp3an1 1449 . . . . . . . . . . . . . 14 ((𝐴 βŠ† ℝ ∧ (vol*β€˜π΄) ∈ ℝ) β†’ (vol*β€˜(𝐴 ∩ (𝑠 βˆ– βˆͺ ran ((,) ∘ 𝑔)))) ∈ ℝ)
358357recnd 11239 . . . . . . . . . . . . 13 ((𝐴 βŠ† ℝ ∧ (vol*β€˜π΄) ∈ ℝ) β†’ (vol*β€˜(𝐴 ∩ (𝑠 βˆ– βˆͺ ran ((,) ∘ 𝑔)))) ∈ β„‚)
359353, 354, 358subadd2d 11587 . . . . . . . . . . . 12 ((𝐴 βŠ† ℝ ∧ (vol*β€˜π΄) ∈ ℝ) β†’ (((vol*β€˜π΄) βˆ’ (vol*β€˜(𝐴 βˆ– (𝑠 βˆ– βˆͺ ran ((,) ∘ 𝑔))))) = (vol*β€˜(𝐴 ∩ (𝑠 βˆ– βˆͺ ran ((,) ∘ 𝑔)))) ↔ ((vol*β€˜(𝐴 ∩ (𝑠 βˆ– βˆͺ ran ((,) ∘ 𝑔)))) + (vol*β€˜(𝐴 βˆ– (𝑠 βˆ– βˆͺ ran ((,) ∘ 𝑔))))) = (vol*β€˜π΄)))
360359ad5antr 733 . . . . . . . . . . 11 (((((((𝐴 βŠ† ℝ ∧ (vol*β€˜π΄) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑒 ∈ ℝ ∧ 𝑒 < (vol*β€˜π΄))) ∧ (𝐴 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ≀ ((vol*β€˜π΄) + 1))) ∧ ((βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) βŠ† βˆͺ ran ((,) ∘ 𝑔) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑔)) ≀ ((vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)) + (((vol*β€˜π΄) βˆ’ 𝑒) / 2)))) ∧ (𝑠 ∈ (Clsdβ€˜(topGenβ€˜ran (,))) ∧ (𝑠 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ ((vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) βˆ’ (((vol*β€˜π΄) βˆ’ 𝑒) / 2)) < (vol*β€˜π‘ )))) β†’ (((vol*β€˜π΄) βˆ’ (vol*β€˜(𝐴 βˆ– (𝑠 βˆ– βˆͺ ran ((,) ∘ 𝑔))))) = (vol*β€˜(𝐴 ∩ (𝑠 βˆ– βˆͺ ran ((,) ∘ 𝑔)))) ↔ ((vol*β€˜(𝐴 ∩ (𝑠 βˆ– βˆͺ ran ((,) ∘ 𝑔)))) + (vol*β€˜(𝐴 βˆ– (𝑠 βˆ– βˆͺ ran ((,) ∘ 𝑔))))) = (vol*β€˜π΄)))
361351, 360mpbird 257 . . . . . . . . . 10 (((((((𝐴 βŠ† ℝ ∧ (vol*β€˜π΄) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑒 ∈ ℝ ∧ 𝑒 < (vol*β€˜π΄))) ∧ (𝐴 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ≀ ((vol*β€˜π΄) + 1))) ∧ ((βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) βŠ† βˆͺ ran ((,) ∘ 𝑔) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑔)) ≀ ((vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)) + (((vol*β€˜π΄) βˆ’ 𝑒) / 2)))) ∧ (𝑠 ∈ (Clsdβ€˜(topGenβ€˜ran (,))) ∧ (𝑠 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ ((vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) βˆ’ (((vol*β€˜π΄) βˆ’ 𝑒) / 2)) < (vol*β€˜π‘ )))) β†’ ((vol*β€˜π΄) βˆ’ (vol*β€˜(𝐴 βˆ– (𝑠 βˆ– βˆͺ ran ((,) ∘ 𝑔))))) = (vol*β€˜(𝐴 ∩ (𝑠 βˆ– βˆͺ ran ((,) ∘ 𝑔)))))
362345, 361eqtr4d 2776 . . . . . . . . 9 (((((((𝐴 βŠ† ℝ ∧ (vol*β€˜π΄) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑒 ∈ ℝ ∧ 𝑒 < (vol*β€˜π΄))) ∧ (𝐴 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ≀ ((vol*β€˜π΄) + 1))) ∧ ((βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) βŠ† βˆͺ ran ((,) ∘ 𝑔) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑔)) ≀ ((vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)) + (((vol*β€˜π΄) βˆ’ 𝑒) / 2)))) ∧ (𝑠 ∈ (Clsdβ€˜(topGenβ€˜ran (,))) ∧ (𝑠 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ ((vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) βˆ’ (((vol*β€˜π΄) βˆ’ 𝑒) / 2)) < (vol*β€˜π‘ )))) β†’ (volβ€˜(𝑠 βˆ– βˆͺ ran ((,) ∘ 𝑔))) = ((vol*β€˜π΄) βˆ’ (vol*β€˜(𝐴 βˆ– (𝑠 βˆ– βˆͺ ran ((,) ∘ 𝑔))))))
363331, 362breqtrrd 5176 . . . . . . . 8 (((((((𝐴 βŠ† ℝ ∧ (vol*β€˜π΄) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑒 ∈ ℝ ∧ 𝑒 < (vol*β€˜π΄))) ∧ (𝐴 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ≀ ((vol*β€˜π΄) + 1))) ∧ ((βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) βŠ† βˆͺ ran ((,) ∘ 𝑔) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑔)) ≀ ((vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)) + (((vol*β€˜π΄) βˆ’ 𝑒) / 2)))) ∧ (𝑠 ∈ (Clsdβ€˜(topGenβ€˜ran (,))) ∧ (𝑠 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ ((vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) βˆ’ (((vol*β€˜π΄) βˆ’ 𝑒) / 2)) < (vol*β€˜π‘ )))) β†’ 𝑒 < (volβ€˜(𝑠 βˆ– βˆͺ ran ((,) ∘ 𝑔))))
364 fvex 6902 . . . . . . . . 9 (volβ€˜(𝑠 βˆ– βˆͺ ran ((,) ∘ 𝑔))) ∈ V
365 eqeq1 2737 . . . . . . . . . . . 12 (𝑣 = (volβ€˜(𝑠 βˆ– βˆͺ ran ((,) ∘ 𝑔))) β†’ (𝑣 = (volβ€˜π‘) ↔ (volβ€˜(𝑠 βˆ– βˆͺ ran ((,) ∘ 𝑔))) = (volβ€˜π‘)))
366365anbi2d 630 . . . . . . . . . . 11 (𝑣 = (volβ€˜(𝑠 βˆ– βˆͺ ran ((,) ∘ 𝑔))) β†’ ((𝑏 βŠ† 𝐴 ∧ 𝑣 = (volβ€˜π‘)) ↔ (𝑏 βŠ† 𝐴 ∧ (volβ€˜(𝑠 βˆ– βˆͺ ran ((,) ∘ 𝑔))) = (volβ€˜π‘))))
367366rexbidv 3179 . . . . . . . . . 10 (𝑣 = (volβ€˜(𝑠 βˆ– βˆͺ ran ((,) ∘ 𝑔))) β†’ (βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑏 βŠ† 𝐴 ∧ 𝑣 = (volβ€˜π‘)) ↔ βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑏 βŠ† 𝐴 ∧ (volβ€˜(𝑠 βˆ– βˆͺ ran ((,) ∘ 𝑔))) = (volβ€˜π‘))))
368 breq2 5152 . . . . . . . . . 10 (𝑣 = (volβ€˜(𝑠 βˆ– βˆͺ ran ((,) ∘ 𝑔))) β†’ (𝑒 < 𝑣 ↔ 𝑒 < (volβ€˜(𝑠 βˆ– βˆͺ ran ((,) ∘ 𝑔)))))
369367, 368anbi12d 632 . . . . . . . . 9 (𝑣 = (volβ€˜(𝑠 βˆ– βˆͺ ran ((,) ∘ 𝑔))) β†’ ((βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑏 βŠ† 𝐴 ∧ 𝑣 = (volβ€˜π‘)) ∧ 𝑒 < 𝑣) ↔ (βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑏 βŠ† 𝐴 ∧ (volβ€˜(𝑠 βˆ– βˆͺ ran ((,) ∘ 𝑔))) = (volβ€˜π‘)) ∧ 𝑒 < (volβ€˜(𝑠 βˆ– βˆͺ ran ((,) ∘ 𝑔))))))
370364, 369spcev 3597 . . . . . . . 8 ((βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑏 βŠ† 𝐴 ∧ (volβ€˜(𝑠 βˆ– βˆͺ ran ((,) ∘ 𝑔))) = (volβ€˜π‘)) ∧ 𝑒 < (volβ€˜(𝑠 βˆ– βˆͺ ran ((,) ∘ 𝑔)))) β†’ βˆƒπ‘£(βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑏 βŠ† 𝐴 ∧ 𝑣 = (volβ€˜π‘)) ∧ 𝑒 < 𝑣))
371196, 363, 370syl2anc 585 . . . . . . 7 (((((((𝐴 βŠ† ℝ ∧ (vol*β€˜π΄) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑒 ∈ ℝ ∧ 𝑒 < (vol*β€˜π΄))) ∧ (𝐴 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ≀ ((vol*β€˜π΄) + 1))) ∧ ((βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) βŠ† βˆͺ ran ((,) ∘ 𝑔) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑔)) ≀ ((vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)) + (((vol*β€˜π΄) βˆ’ 𝑒) / 2)))) ∧ (𝑠 ∈ (Clsdβ€˜(topGenβ€˜ran (,))) ∧ (𝑠 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ ((vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) βˆ’ (((vol*β€˜π΄) βˆ’ 𝑒) / 2)) < (vol*β€˜π‘ )))) β†’ βˆƒπ‘£(βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑏 βŠ† 𝐴 ∧ 𝑣 = (volβ€˜π‘)) ∧ 𝑒 < 𝑣))
372163, 371rexlimddv 3162 . . . . . 6 ((((((𝐴 βŠ† ℝ ∧ (vol*β€˜π΄) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑒 ∈ ℝ ∧ 𝑒 < (vol*β€˜π΄))) ∧ (𝐴 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ≀ ((vol*β€˜π΄) + 1))) ∧ ((βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴) βŠ† βˆͺ ran ((,) ∘ 𝑔) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑔)) ≀ ((vol*β€˜(βˆͺ ran ((,) ∘ 𝑓) βˆ– 𝐴)) + (((vol*β€˜π΄) βˆ’ 𝑒) / 2)))) β†’ βˆƒπ‘£(βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑏 βŠ† 𝐴 ∧ 𝑣 = (volβ€˜π‘)) ∧ 𝑒 < 𝑣))
373142, 372exlimddv 1939 . . . . 5 (((((𝐴 βŠ† ℝ ∧ (vol*β€˜π΄) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑒 ∈ ℝ ∧ 𝑒 < (vol*β€˜π΄))) ∧ (𝐴 βŠ† βˆͺ ran ((,) ∘ 𝑓) ∧ (vol*β€˜βˆͺ ran ((,) ∘ 𝑓)) ≀ ((vol*β€˜π΄) + 1))) β†’ βˆƒπ‘£(βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑏 βŠ† 𝐴 ∧ 𝑣 = (volβ€˜π‘)) ∧ 𝑒 < 𝑣))
37478, 373exlimddv 1939 . . . 4 ((((𝐴 βŠ† ℝ ∧ (vol*β€˜π΄) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑒 ∈ ℝ ∧ 𝑒 < (vol*β€˜π΄))) β†’ βˆƒπ‘£(βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑏 βŠ† 𝐴 ∧ 𝑣 = (volβ€˜π‘)) ∧ 𝑒 < 𝑣))
375 eqeq1 2737 . . . . . . 7 (𝑦 = 𝑣 β†’ (𝑦 = (volβ€˜π‘) ↔ 𝑣 = (volβ€˜π‘)))
376375anbi2d 630 . . . . . 6 (𝑦 = 𝑣 β†’ ((𝑏 βŠ† 𝐴 ∧ 𝑦 = (volβ€˜π‘)) ↔ (𝑏 βŠ† 𝐴 ∧ 𝑣 = (volβ€˜π‘))))
377376rexbidv 3179 . . . . 5 (𝑦 = 𝑣 β†’ (βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑏 βŠ† 𝐴 ∧ 𝑦 = (volβ€˜π‘)) ↔ βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑏 βŠ† 𝐴 ∧ 𝑣 = (volβ€˜π‘))))
378377rexab 3690 . . . 4 (βˆƒπ‘£ ∈ {𝑦 ∣ βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑏 βŠ† 𝐴 ∧ 𝑦 = (volβ€˜π‘))}𝑒 < 𝑣 ↔ βˆƒπ‘£(βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑏 βŠ† 𝐴 ∧ 𝑣 = (volβ€˜π‘)) ∧ 𝑒 < 𝑣))
379374, 378sylibr 233 . . 3 ((((𝐴 βŠ† ℝ ∧ (vol*β€˜π΄) ∈ ℝ) ∧ 𝐴 ∈ dom vol) ∧ (𝑒 ∈ ℝ ∧ 𝑒 < (vol*β€˜π΄))) β†’ βˆƒπ‘£ ∈ {𝑦 ∣ βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑏 βŠ† 𝐴 ∧ 𝑦 = (volβ€˜π‘))}𝑒 < 𝑣)
3802, 3, 42, 379eqsupd 9449 . 2 (((𝐴 βŠ† ℝ ∧ (vol*β€˜π΄) ∈ ℝ) ∧ 𝐴 ∈ dom vol) β†’ sup({𝑦 ∣ βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑏 βŠ† 𝐴 ∧ 𝑦 = (volβ€˜π‘))}, ℝ, < ) = (vol*β€˜π΄))
381380eqcomd 2739 1 (((𝐴 βŠ† ℝ ∧ (vol*β€˜π΄) ∈ ℝ) ∧ 𝐴 ∈ dom vol) β†’ (vol*β€˜π΄) = sup({𝑦 ∣ βˆƒπ‘ ∈ (Clsdβ€˜(topGenβ€˜ran (,)))(𝑏 βŠ† 𝐴 ∧ 𝑦 = (volβ€˜π‘))}, ℝ, < ))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 397   = wceq 1542  βˆƒwex 1782   ∈ wcel 2107  {cab 2710  βˆƒwrex 3071   βˆ– cdif 3945   βˆͺ cun 3946   ∩ cin 3947   βŠ† wss 3948  βˆͺ cuni 4908   class class class wbr 5148   Or wor 5587   Γ— cxp 5674  dom cdm 5676  ran crn 5677   ∘ ccom 5680  βŸΆwf 6537  β€˜cfv 6541  (class class class)co 7406   ↑m cmap 8817  supcsup 9432  β„‚cc 11105  β„cr 11106  0cc0 11107  1c1 11108   + caddc 11110  +∞cpnf 11242  β„*cxr 11244   < clt 11245   ≀ cle 11246   βˆ’ cmin 11441   / cdiv 11868  β„•cn 12209  2c2 12264  β„+crp 12971  (,)cioo 13321  [,)cico 13323  seqcseq 13963  abscabs 15178  topGenctg 17380  Topctop 22387  TopBasesctb 22440  Clsdccld 22512  vol*covol 24971  volcvol 24972
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7722  ax-inf2 9633  ax-cnex 11163  ax-resscn 11164  ax-1cn 11165  ax-icn 11166  ax-addcl 11167  ax-addrcl 11168  ax-mulcl 11169  ax-mulrcl 11170  ax-mulcom 11171  ax-addass 11172  ax-mulass 11173  ax-distr 11174  ax-i2m1 11175  ax-1ne0 11176  ax-1rid 11177  ax-rnegex 11178  ax-rrecex 11179  ax-cnre 11180  ax-pre-lttri 11181  ax-pre-lttrn 11182  ax-pre-ltadd 11183  ax-pre-mulgt0 11184  ax-pre-sup 11185
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-iin 5000  df-disj 5114  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-se 5632  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6298  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6493  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-isom 6550  df-riota 7362  df-ov 7409  df-oprab 7410  df-mpo 7411  df-of 7667  df-om 7853  df-1st 7972  df-2nd 7973  df-frecs 8263  df-wrecs 8294  df-recs 8368  df-rdg 8407  df-1o 8463  df-2o 8464  df-oadd 8467  df-omul 8468  df-er 8700  df-map 8819  df-pm 8820  df-en 8937  df-dom 8938  df-sdom 8939  df-fin 8940  df-fi 9403  df-sup 9434  df-inf 9435  df-oi 9502  df-dju 9893  df-card 9931  df-acn 9934  df-pnf 11247  df-mnf 11248  df-xr 11249  df-ltxr 11250  df-le 11251  df-sub 11443  df-neg 11444  df-div 11869  df-nn 12210  df-2 12272  df-3 12273  df-4 12274  df-n0 12470  df-z 12556  df-uz 12820  df-q 12930  df-rp 12972  df-xneg 13089  df-xadd 13090  df-xmul 13091  df-ioo 13325  df-ico 13327  df-icc 13328  df-fz 13482  df-fzo 13625  df-fl 13754  df-seq 13964  df-exp 14025  df-hash 14288  df-cj 15043  df-re 15044  df-im 15045  df-sqrt 15179  df-abs 15180  df-clim 15429  df-rlim 15430  df-sum 15630  df-rest 17365  df-topgen 17386  df-psmet 20929  df-xmet 20930  df-met 20931  df-bl 20932  df-mopn 20933  df-top 22388  df-topon 22405  df-bases 22441  df-cld 22515  df-cmp 22883  df-conn 22908  df-ovol 24973  df-vol 24974
This theorem is referenced by:  ismblfin  36518
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